The tab appearing second from the left on your Twitter homepage contains your @UserName. Clicking on it calls up any activity occurring on Twitter that involves you or your user name.
When you click that tab, in the middle column (where your home timeline normally appears) you will now see other stuff about you. For example, any recent retweets of your messages may appear, along with a list of who's followed you recently.
You should also see any messages sent to you directly, such as mentions of your user name or @replies. And if anyone has favorited your messages ("favorite" on Twitter it's similar to "like" on Facebook) that should show up, too.
Once you get a lot of Twitter followers and conversations going, you may want to cut through the clutter and simply see who has mentioned you on Twitter, which is the equivalent of someone sending you a direct message. To do that, check the small "show mentions only" box at the top of your message stream. Then the timeline of stuff about you will change; it shows what used to be the @Mention tab on your Twitter homepage, namely,just your @mentions.
The next tab, Activity, is less narcissistic and a tad more complicated.