What exactly does a non-parametric test accomplish & What do you do with the results? I have a feeling this may have been asked elsewhere, but not really with the type of basic description I need.  I know non-parametric relies on the median instead of the mean to compare... something.  I also believe it relies on "degrees of freedom"(?) instead of standard deviation.  Correct me if I'm wrong, though.
I've done pretty good research, or so I'd thought, trying to understand the concept, what the workings are behind it, what the test results really mean, and/or what to even do with the test results; however, no one seems to ever venture into that area.
For the sake of simplicity let's stick with the Mann-Whitney U-test, which I've noticed is quite popular (and also seemingly misused and overused too in order to force one's "square model into a circle hole").  If you'd like to describe the other tests as well feel free, although I feel once I understand one, I can understand the others in an analogous way towards various t-tests etc.
Let's say I run a non-parametric test with my data and I get this result back:
2 Sample Mann-Whitney - Customer Type       

Test Information        
H0: Median Difference = 0       
Ha: Median Difference ≠ 0       

Size of Customer    Large   Small
Count                    45    55
Median                    2     2

Mann-Whitney Statistic: 2162.00 
p-value (2-sided, adjusted for ties):   0.4156  

I'm familiar with other methods, but what is different here?  Should we want the p-value to be lower than .05?  What does the "Mann-Whitney statistic" mean?  Is there any use for it?  Does this information here just verify or not verify that a particular source of data I have should or should not be used?
I have a reasonable amount of experience with regression and the basics, but am very curious about this "special" non-parametric stuff - which I know will have it's own shortcomings.
Just imagine I'm a fifth grader and see if you can explain it to me.
 A: You asked to be corrected if wrong. Here are some comments under that heading to complement @Peter Flom's positive suggestions. 


*

*"non-parametric relies on the median instead of the mean": often in practice, but that's not a definition. Several non-parametric tests (e.g. chi-square) have nothing to do with medians. 

*relies on degrees of freedom instead of standard deviation; that's very confused. The idea of degrees of freedom is in no sense an alternative to standard deviation; degrees of freedom as an idea applies right across statistics. 

*"a particular source of data I have should or should not be used": this question has nothing to do with the significance test you applied, which is just about the difference between subsets of data and is phrased in terms of difference between medians. 
A: 
I know non-parametric relies on the median instead of the mean

Hardly any nonparametric tests actually "rely on" medians in this sense. I can only think of a couple... and the only one I expect you'd be likely to have even heard of would be the sign test.

to compare...something.

If they relied on medians, presumably it would be to compare medians. But - in spite of what a number of sources try to tell you - tests like the signed rank test, or the Wilcoxon-Mann-Whitney or the Kruskal-Wallis are not really a test of medians at all; if you make some additional assumptions, you can regard the Wilcoxon-Mann-Whitney and the Kruskal-Wallis as tests of medians, but under the same assumptions (as long as the distributional means exist) you could equally regard them as a test of means.
The actual location-estimate relevant to the Signed Rank test is the median of pairwise averages within-sample (over $\frac12 n(n+1)$ pairs including self-pairs), the one for the Wilcoxon-Mann-Whitney is the median of pairwise differences across-samples.

I also believe it relies on "degrees of freedom?" instead of standard deviation. Correct me if I'm wrong though.

Most nonparametric tests don't have 'degrees of freedom' in the specific sense that the chi-squared or the t-test of the F-test do (each of which has to do with the number of degrees of freedom in an estimate of variance), though the distribution of many change with sample size and you might regard that as somewhat akin to degrees of freedom in the sense that the tables change with sample size. The samples do of course retain their properties and have n degrees of freedom in that sense but the degrees of freedom in the distribution of a test statistic is not typically something we're concerned with. It can happen that you have something more like degrees of freedom - for example, you could certainly make an argument that the Kruskal-Wallis does have degrees of freedom in basically the same sense that a chi-square does, but it's usually not looked at that way (for example, if someone's talking about the degrees of freedom of a Kruskal-Wallis, they will nearly always mean the d.f. of the chi-square approximation to the distribution of the statistic).
A good discussion of degrees of freedom may be found here/

I've done pretty good research, or so I've thought, trying to understand the concept, what the workings are behind it, what the test results really mean, and/or what to even do with the test results; however no one seems to ever venture into that area.

I'm not sure what you mean by this.
I could suggest some books, like Conover's Practical Nonparametric Statistics, and if you can get it, Neave and Worthington's book (Distribution-Free Tests), but there are
many others - Marascuilo & McSweeney, Hollander & Wolfe, or Daniel's book for example. I suggest you read at least 3 or 4 of the ones that speak to you best, preferably ones that explain things as differently as possible (this would mean at least reading a little of perhaps 6 or 7 books to find say 3 that suit).

For the sake of simplicity lets stick with the Mann Whitney U test, which I've noticed is quite popular

It is, which is what puzzled me about your statement "no one seems to ever venture into that area" - many people who use these tests do 'venture into the area' you were talking about.

- and also seemingly misused and overused

I'd say nonparametric tests are generally underused if anything (including the Wilcoxon-Mann-Whitney) -- most especially permutation/randomization tests, though I wouldn't necessarily dispute that they're frequently misused (but so are parametric tests, even more so).

Let's say I run a non-parametric test with my data and I get this result back:

[snip...]

I'm familiar with other methods, but what is different here?

Which other methods do you mean? What do you want me to compare this to?
Edit: You mention regression later; I assume then that you are familiar with a two-sample t-test (since it's really a special case of regression).
Under the assumptions for the ordinary two-sample t-test, the null hypothesis has that the two populations are identical, against the alternative that one of the distributions has shifted. If you look at the first of the two sets of hypotheses for the Wilcoxon-Mann-Whitney below, the basic thing being tested there is almost identical; it's just that the t-test is based on assuming the samples come from identical normal distributions (apart from possible location-shift). If the null hypothesis is true, and the accompanying assumptions are true, the test statistic has a t-distribution. If the alternative hypothesis is true, then the test-statistic becomes more likely to take values that don't look consistent with the null hypothesis but do look consistent with the alternative -- we focus on the most unusual, or extreme outcomes (the ones most consistent with the alternative) - if they occur, we conclude that the samples we got would not have occurred by chance when the null was true (they could do, but the probability of a result at least that much consistent with the alternative is so low that we consider the alternative hypothesis a better explanation for what we observe than "the null hypothesis along with the operation of chance").
The situation is very similar with the Wilcoxon-Mann-Whitney, but it measures the deviation from the null somewhat differently. In fact, when the assumptions of the t-test are true*, it's almost as good as the best possible test (which is the t-test).
*(which in practice is never, though that's not really as much of a problem as it sounds)

Indeed, it's possible to consider the Wilcoxon-Mann-Whitney as effectively a "t-test" performed on the ranks of the data - though then it doesn't have a t-distribution; the statistic is a monotonic function of a two-sample t-statistic computed on the ranks of the data, so it induces the same ordering on the sample space (that is a "t-test" on the ranks - appropriately performed - would generate the same p-values as a Wilcoxon-Mann-Whitney), so it rejects exactly the same cases.
[You'd think that just using the ranks would be throwing away a lot of information, but when the data are drawn from normal populations with the same variance, almost all the information about location-shift is in the patterns of the ranks. The actual data values (conditional on their ranks) add very little additional information to that. If you go heavier-tailed than normal, it's not long before the Wilcoxon-Mann-Whitney test has better power, as well as retaining its nominal significance level, so that 'extra' information above the ranks eventually becomes not just uninformative but in some sense, misleading. However, near-symmetric heavy-tailedness is a rare situation; what you often see in practice is skewness.]
The basic ideas are quite similar, the p-values have the same interpretation (the probability of a result as, or more extreme, if the null hypothesis were true) -- right down to the interpretation of a location-shift, if you make the requisite assumptions (see the discussion of the hypotheses near the end of this post).
If I did the same simulation as in the plots above for the t-test, the plots would look very similar - the scale on the x- and y-axes would look different, but the basic appearance would be similar.

Should we want the p-value to be lower than .05?

You shouldn't "want" anything there. The idea is to find out if the samples are more different (in a location-sense) than can be explained by chance, not to 'wish' a particular outcome.
If I say "Can you go see what color Raj's car is please?", if I want an unbiased assessment of it I don't want you to be going "Man, I really, really hope it's blue! It just has to be blue". Best to just see what the situation is, rather than to go in with some 'I need it to be something'.
If your chosen significance level is 0.05, then you'll reject the null hypothesis when the p-value is ≤ 0.05. But failure to reject when you have a big enough sample size to nearly always detect relevant effect-sizes is at least as interesting, because it says that any differences that exist are small.

What does the "mann whitley" number mean?

The Mann-Whitney statistic.
It's really only meaningful in comparison with the distribution of values it can take when the null hypothesis is true (see the above diagram), and that depends on which of several particular definitions any particular program might use.

Is there any use for it?

Usually you don't care about the exact value as such, but where it lies in the null-distribution (whether it's more or less typical of the values you should see when the null hypothesis is true, or whether it's more extreme)
(Edit: You can obtain or work out some directly informative quantities when doing such a test - like the location shift or $P(X<Y)$ discussed below, and indeed you can work out the second one fairly directly from the statistic, but the statistic alone isn't a very informative number)

Does this data here just verify or not verify that a particular source of data I have should or should not be used?

This test doesn't say anything about "a particular source of data I have should or should not be used".
See my discussion of the two ways of looking at the WMW hypotheses below.

I have a reasonable amount of experience with regression and the basics, but am very curious about this "special" non-parametric stuff

There's nothing particularly special about nonparametric tests (I'd say the 'standard' ones are in many ways even more basic than the typical parametric tests) -- as long as you actually understand hypothesis testing.
That's probably a topic for another question, however.

There are two main ways to look at the Wilcoxon-Mann-Whitney hypothesis test.
i) One is to say "I'm interested in location-shift - that is that under the null hypothesis, the two populations have the same (continuous) distribution, against the alternative that one is 'shifted' up or down relative to the other"
The Wilcoxon-Mann-Whitney works very well if you make this assumption (that your alternative is just a location shift)
In this case, the Wilcoxon-Mann-Whitney actually is a test for medians ... but equally it's a test for means, or indeed any other location-equivariant statistic (90th percentiles, for example, or trimmed means, or any number of other things), since they're all affected the same way by location-shift.
The nice thing about this is that it's very easily interpretable -- and it's easy to generate a confidence interval for this location-shift.

However, the Wilcoxon-Mann-Whitney test is sensitive to other kinds of difference than a location shift.
ii) The other is to take the fully general approach. You can characterize this as a test for the probability that a random value from population 1 is less than a random value from population 2 (and indeed, you can turn your Wilcoxon-Mann-Whitney statistic into a direct estimate of that probability, if you're so inclined; the Mann&Whitney formulation in terms of U-statistics counts the number of times one exceeds the other in the samples, you only need scale that to achieve an estimate of the probability); the null is that the population probability is $\frac{1}{2}$, against the alternative that it differs from $\frac{1}{2}$.

However, while it can work okay in this situation, the test is formulated on the assumption of exchangability under the null. Among other things that would require that in the null case the two distributions are the same. If we don't have that, and are instead are in a slightly different situation like the one pictured above, we won't typically have a test with significance level $\alpha$. In the pictured case it would likely be a bit lower.
So while it "works" in the sense that it tends not to reject when $H_0$ is true and tends to reject more when $H_0$ is false, you want the distributions to be pretty close to identical under the null or the test doesn't behave the way we would expect it to.
A: You "want" the same things from a p-value here that you want in any other test.
The U statistic is the result of a calculation, just like the t statistic, the odds ratio, the F statistic, or what have you. The formula can be found lots of places. It's not very intuitive, but then, neither are other test statistics until you get used to them (we recognize a t of 2 as being in the significant range because we see them all the time). 
The rest of the output in your block text should be clear.
For a more general introduction to nonparametric tests, I echo @NickCox .... get a good book. Non-parametric simply means "without parameters"; there are many non-parametric tests and statistics for a wide variety of purposes. 
A: Suppose you and I are coaching track teams. Our athletes come from the same school, are similar ages, and the same gender (i.e., they're drawn from the same population), but I claim to have discovered a Revolutionary New Training System that will make my team members run much faster than yours. How can I convince you that it really does work?
We have a race. 
Afterward, I sit down and compute the average time for the members of my team and the average time for the members of yours. I'll claim victory if the mean time for my athletes is not only faster than the mean for yours, but the difference is also large compared to the "scatter", or standard deviation, of our results. 

This is essentially a [$t$-test][1]. We're assuming that the data arises from distributions with specific parameters, in this case a mean and standard deviation. The test estimates those parameters and compares one of them (the mean). It is, consequently, called a parametric test, since we are comparing these parameters.


"But Matt", you complain, "this isn't quite fair. Our teams are pretty similar, but you--due to pure chance--ended up with the fastest runner in the district. He's not in the same league as everyone else; he's practically a freak of Nature. He finished 3 minutes before the next-fastest finisher, which reduces your average time a lot, but the rest of the competitors are pretty evenly mixed. Let's look at the finish order instead. If your method really works, the earlier finishers should mostly be from your team, but if it doesn't the finish order should be pretty random. This doesn't give undue weight to your super-star!"

This method is essentially the [Mann-Whitney U Test][2] (also called the Wilcoxon Rank Sum Test, Manning-Whitney-Wilcoxon Test, and several other permutations besides!). Note that unlike the $t$-test, we're not assuming that the data comes from specific distributions, nor are we computing any parameters for them. Instead, we're comparing the relative ranks of the data points directly. 
That's the major distinction--parametric tests model things with distributions and compare the parameters of these distributions; non-parametric tests....don't and operate more directly on the data. As with parametric tests, non-parametric test statistics are also constructed so that the $p$ values are uniformly distributed on [0,1] under the null hypothesis and clustered towards 0 in the presence of an effect. You would report and interpret them just like the results of a parametric test. 
I'm not sure about the relative popularity of parametric and non-parametric methods. Some non-parametric methods (e.g., histograms!) are in nearly universal use; others might be over or under-used. I suspect that the Mann-Whitney U Test ought to be used more, rather than less frequently. It's about as efficient as a $t$-test on normally distributed data and actually does better than the $t$-test on sufficiently non-normal data. It's also fairly robust to outliers. Plus, you can use it on ordinal data too (e.g., finish order rather than just finish time), which makes it more broadly applicable than a $t$-test.
A: As a response to a recently closed question, this addresses the above as well. Below is a quote from Bradley's classic Distribution-Free Statistical Tests (1968, p. 15–16) which, while a bit long, is a pretty clear explanation, I believe.

The terms nonparametric and distribution-free are not synonymous, and
  neitherterm provides an entirely satisfactory description of the class
  of statistics to which they are intended to refer.…Roughly speaking, a
  nonparametric test is one which makes no hypothesis about the value of
  a parameter in a statistical density function, whereas a
  distribution-free test is one which makes no assumptions about the
  precise form of the sampled population. The definitions are not
  mutually exclusive, and a test can be both distribution-free and
  parametric.…In order to be entirely clear about what is meant by
  distribution-free, it is necessary to distinguish between three
  distributions: (a) that of the sampled population; (b) that of the
  observation-characteristic actually used by the test; and (c) that of
  the test statistic. The distribution from which the tests are "free"
  is that of (a), the sampled population. And the freedom they enjoy is
  usually relative.…However the assumptions are never so elaborate as to
  imply a population whose distribution is completely specified.…The
  reason…is very simple: the magnitudes are not used as such in the
  [nonparametric] test, nor is any other strongly-linked population
  attribute of the variate. Instead sample-linked charachteristics of
  the obtained observations…provide the informatikon used by the test
  statistic.…Thus while both parametric and nonparametric tests require
  that the form f a distribution, associated with observations, be
  fully known, that knowledge, in the parametric case, is generally not
  forthcoming ad the required distribution of magnitudes must
  therefore be "assumed" or inferred on the basis of approximate or
  incomplete information. In the nonparametric case, on the other and,
  the distrbution of the observation characteristic is usually known
  precisely from a priori considerations and need not, therefore, be
  "assumed." The difference, then, is not one of requirement but rather
  of what is required and of certainty that the requirement will be met.

