When do Post-hoc tests require an adjustment for multiple comparisons What happens to the Type 1 error rate, the Type 2 error rate, and Power if you do 
not adjust for multiple comparisons when you should?  What happens to these as a result 
of using something like Bonferroni’s adjustment?
 A: If you do a bunch of tests, the chances of at least one Type I error (when the null hypothesis is true) will go up, sometimes dramatically.
A Bonferroni adjustment (as with most other multiple comparison procedures) is an attempt to hold the overall Type I error rate from the collection of pairwise comparisons to no more than a prespecified rate of Type I errors in total.
So if you don't do some such adjustment, you'll end up with a higher overall Type I error rate than the per-test error rate, so if the per-test error rate is your desired overall Type I rate, you'll obviously be taking a risk of more Type I errors; as such the Type II error rate will be lower (when the null is actually false), relative to when you do make the adjustment.
Since power is simply the complement of the Type II error rate, power goes down when you make the adjustment (or, by comparison is higher when you don't).
(What's worth noting in passing, perhaps, is that the bad side of not adjusting only occurs when the null is true for those comparisons; it would be rare indeed for this to be the case. This is part of the reason I think people worry too much about tight control of familywise error. In a large collection of tests, depending on the relative costs of the two types of error, then I'm often content if there's a Type I error in there in any case. On the other hand, if you don't want to reject when the null is very nearly true, you may want to worry about controlling it more.)
A: In addition to @Glen_b 's excellent answer (+1) I'd add
1) "Familywise" begs the question of what a family is. All the analyses in one paper? All the analyses on one data set? All the analyses related to one question? All the analyses you do in your life? What about analyses that other people do on the same data?
2) In addition, we default to "5%" and "20%" for type I and type II, but while we treat these as almost sacred, there's no reason to do so. Sometimes a type I error is bad; sometimes it isn't.  (But try convincing a journal editor of that!) 
A: @Glen_b & @Peter Flom have provided good answers.  Let me add one more detail:  
It is not simply that we lose power when we make an adjustment to control the familywise type I error rate.  The existence of multiple comparisons means that power is already compromised.  
Discussions of the problem of multiple comparisons typically center around th familywise type I error rate.  We can imagine an analogous familywise power rate.  Consider a situation in which your family contains two independent analyses, where both nulls are false, and that the power to reject them is 80% in both cases.  Given this setup, the power to reject both false nulls is 64%, even if the familywise type I error rate is completely ignored / you don't do anything to address it.  
Discussions of familywise type I error rates in multiple comparison situations without accompanying discussions like this are another manifestation of the traditional (& dogmatic) concern with only the possibility of / negatives associated with type I errors in hypothesis testing.  
