I think it is hard to explain this topic in just a few words. As you already mentioned, all of them are diagnostics to detect outliers. Whereas Mahalanobis and Leverage only take account for the IV side, Cook's Di analyzes the discrepancies on the DV side.

Mahalanobis and Leverage are very similar and their logic is to estimate the distance of a specific value Xi from the according centroid (i.e. the mean value). This can of course be generalized and used with more than one IV. The bigger the value, the more extreme is this case. For Mahalanobis it is also possible to do some kind of "significance" test, using a chi square distribution.

Cook*s distance on the other hand investigates changes in the predicted Y scores. To put it simple: it is the squared and summed difference between predicted Y and predicted Y if case i is eleminated. This whole term is standardized (i.e. divided) by the mean squared residuals. So it gives an estimate how much the predicted Y will change, if one case is eleminated and the bigger the worse. As a rule of thumb, values >1 give rise for concern.

I hope this explanation helps a bit. But please have a look at the following books, which explain the topic quite good with small examples, as far as I remember:

Cohen, Cohen, West, & Aiken - Applied Multiple Regression/Correlation Analysis

Stevens - Applied Multivariate Statistics for the Social Sciences

Field - Discovering Statistics Using SPSS

What's so difficult in the Wikipedia articles?

What books/resources about that topic have you read?