People always have a goal in mind when they make a character. For many, it's a very broad goal, such as leveling a DK to be their tanking character or trying out a mage. Many people, on the other hand, create characters for a much more specific purpose. A common example is the ubiquitous bank alt, a low-level character stowed in a major city whose entire life consists of nothing but looking good while carrying huge loads from the AH to the nearest mailbox. There have been several pacifist characters, at least one person who leveled to the cap completely naked, and one who leveled to cap without dying once.
I recently made a concept character. I have a paladin, death knight, and warrior all at level 85, all specced and geared to tank. I decided late in Wrath that I would roll a worgen druid in order to complete the set. As Cataclysm came closer, I decided that I would work to experience the full spectrum of bear tanking. So I resolved to level my druid without any heirlooms, using only gear that I gain through questing and running dungeons. I also resolved to run every 5-man dungeon at least once while at the appropriate level. He's now frolicking through the snows of Northrend at level 72, has tanked every 5-man instance from Deadmines to UBRS, from Ramparts to Magister's Terrace, and is working his way through the 5-mans of Wrath. It's been a blast, especially since the buff to Swipe.
Have you ever made a character just to play the game with your own twist?