An ad hominem fallacy occurs when someone attempts to refute an argument by attacking the person making the argument, rather than the argument itself. For instance:

“My opponent claims that the Earth is round, but he’s also a convicted Horse-Shaver, so we can’t very well trust anything he has to say. Therefore, the Earth is flat.”

Or, more generally:

“My opponent says that A is the case.
But my opponent has unpleasant quality Y.
Therefore A is false.”

This fallacy is extremely popular in politics and frat house arguments. (Which might be a distinction without a difference.)

For more excellent information about what is or isn’t an ad hominem, check out this article by Stephen Bond. Bond makes the excellent point (which I will now forever onward call Bond’s Law) that a mere insult is not sufficient for an ad hominem fallacy. Rather, the insult must be used as evidence that a particular argument is incorrect.