Alt is simply short for Alternative character, or something similar. It's generally an MMO term, where one player can have potentially dozens of characters. In these cases they are usually divided into one or a few main characters, and all of the rest being alts. Players will play their main characters most of the time, and play an alt when they're bored, or they've maxed out their mains, or their favorite server is down and they cannot access their mains.
It's also an easy way of saying a lower-level character, so to say you're switching to an alt is to say you're going to a lower level.