A Twitter handle is the "at" symbol plus the Twitter user's username. Oddly enough, Twitter doesn't call it a handle; that word's not even in Twitter's glossary, which might confuse new users looking for its definition there.
Usernames can be up to 15 characters long, and Twitter users aren't stuck with them for all eternity. They can change these personal identifiers any time by clicking the Profiles and Settings icon, then Settings. From the username field, they can modify their username. There are no restrictions around what users can call themselves, so it's common for people to be creative to stand out and be memorable.