A mule is generally a secondary character used by a player in order to store more items than one alone can store in his or hers inventory and stash.
This is a huge part of the Diablo II gameplay, where each character have a very small amount of space to store items. While it might be improved upon in a future patch by Blizzard, it's currently a big problem for many players, and a source of some annoyance.
Diablo III will feature a much larger stash with some sort of way to share loot between characters of one player, and will most likely have a lot fewer mules.