What is Cronbach's Alpha intuitively? I'm trying to understand Cronbach's Alpha intuitively. What is the general idea behind this construct? What properties were they trying to ensure that it had?
 A: You can see what it means by studying the formula:
$$
\alpha = \frac{K}{K-1}\left(1-\frac{\sum \sigma^2_{x_i}}{\sigma^2_T}\right)
$$
where  $T=x_1 + x_2 + ... x_K$. $T$ is the total score of a test with $K$ items, each scores $x_i$, respectively. 
Unpack the formula, using what we know about the covariance of a sum of RV's.


*

*If the test items are independent (think $K$ random, Trivial Pursuit questions), then the variance of $T$ is the sum of the variances of the $x_i$ and $\alpha=0$.

*Suppose that the $x_i$ are actually the same question repeated $K$ times. Then $\sigma^2_T=K^2 \sigma^2_x$, and a little algebra shows that $\alpha=1$.


These are the extreme cases. Normally, there will be some positive correlations between the items (assuming that everything is coded in the same direction), so the ratio of the variances will be smaller than 1. The greater the covariances, the larger the value of $\alpha$. 
Remember that there are $K(K-1)/2$ covariances in the $x_i$ to get the variance of $T$, so you need for most things to be reasonably correlated with most other variables to get a healthy $\alpha$. It is, as @ttnphns pointed out, an almost normalized average covariance.
$$
\sigma^2_T = \sum \sigma^2_{x_i} + 2 \sum_{i < j}^K {\rm cov}(x_i,x_j)
$$
This term is in the numerator of the ratio of the variances, so the larger it gets, the smaller that ratio becomes, and the quantity gets closer to 1.
So what does this imply? Take a very simple testing situation, where each item is correlated with an underlying factor with the same loading, thusly:
$$ x_i = \lambda \xi + \epsilon_i$$
Then the covariances are of the form $\lambda^2$. If $\lambda$ is fairly large, relative to the noise $\epsilon$, I'm going to get something close to 1. In fact, if we standardize so that $\sigma^2_x=1$
$$
\alpha = \frac{K}{K-1}\left(1-\frac{1}{1+(K-1)\lambda^2}\right) 
$$
and $\alpha$ is basically a monotone, if non-linear, version of the factor loading.
Sadly, the converse is not true, and large $\alpha$ values can be obtained from a variety of factor structures, or really none at all. The items need to be correlated, on average, but that's not actually saying much. The Cronbach alpha is a test statistic that gets way too much publicity, in my opinion, for what it's worth. Nowadays, there is no reason not to do a factor analysis and confirm whether the test items are performing as one believes they should.
The following graph shows the value of $\alpha$ when there are 20 items with identical loadings, as above.

Psychologists like to get an $\alpha$ greater than 0.80, but that is achievable with a loading of 0.5 -- not exactly a tight test item.
