The at (@) symbol precedes a user's Twitter name as a wee lil' addressing tool, called an "at reply" or @ reply. It lets you speak directly to another user, albeit publicly, regardless of follower/following status. If you want to chat privately, with an extra 20 characters, you can direct message, or DM, each other.
@buddyomine Are you at the show yet? Will you save me a seat? I'm cruising the parking lot, looking, looking. See you in a minute. #CantWait
Please note, if you start a Tweet with an @ reply, that Tweet will be sent ONLY to that account. People who happen to visit your Twitter profile will be able to see that Tweet as part of your public record, but it will not be shown in your other followers' streams. For that reason, you may see periods (.) or open quotes (") in front of @replies that start Tweets. These people want the account owner, and everybody else, to see what they've got to say. There's sense of "calling someone out" in this gesture, for praise or otherwise.