A mule (also known as a bank character, bank toon, bank alt, or banker) is basically a mobile vault, a character that was created for additional item storage capacity. This type of character rarely gets played other than as a gopher between the character needing storage space and the merchants, supplies, and/or product(s), as in the case of a crafter, for instance. Other items typically stored on a mule are reputation tokens, consumables, cloth, and any other non-soulbound items of interest (e.g., [Snowballs]). Before patch 2.3, most guild banks were mules.
A mule is often positioned next to the auction house so its main can mail auctionable items to it. The mule runs between the mailbox and auction house selling goods and funneling the money back to the main.
While mules are usually not developed into viable trade skill alts, enchanting was at one point useful for the conversion of unsalable items into raw materials. However, with the patch 2.0, there is now a minimum skill requirement for disenchanting items, making it much less useful to a low-level mule. Higher level mules with trained enchanting skill (e.g., characters no longer actively played, but still used for storage / disenchanting) are of course still viable, but likely to be far less common than the pre-2.0 level 5 disenchant mule.
With the addition of guild banks in patch 2.3, some players have formed single-member guilds on their mules to gain accesses to more (and in some cases cheaper) storage.