Twitter can look a bit clubby to newbies. One reason is that people on Twitter love to talk about Twitter. Another is that the service has adopted cryptic typographical conventions to signal certain kinds of conversations, and figuring out what they mean is not intuitive.
The most common is the @ sign. In tweets, this usually precedes a username, and it basically means you're talking to or about someone. To reply to a tweet from me, you'd begin your post with @fmanjoo. (This doesn't mean that only I see you post—your reply to me is public.)
Another common symbol is the hash tag, #. People use this to say that their tweets are about a specific subject. For instance, when folks were talking about the terrorist attacks in India last year, they included the #mumbai tag in their tweets so others could follow the conversation. You can find these conversations through Twitter's search engine. Type in #sxsw and you'll find all the messages about the South by Southwest festival.
One other convention: RT means "re-tweet." Starting a tweet with RT means you're echoing someone else's post to your group of followers. For example, if you liked my colleague John Swansburg's post about The Cosby Show, you'd write, "RT @swansburg A high water mark of 1980s situation comedy: http://bit.ly/3NBdl7 Nay, situation comedy, period."