Twitter is a service for friends, family, and co–workers to communicate and stay connected through the exchange of quick, frequent messages. People write short updates, often called "tweets" of 140 characters or fewer. These messages are posted to your profile or your blog, sent to your followers, and are searchable on Twitter search.
When I first used the service a few months back, I really couldn't make heads or tales of it. I just don't have all that much to "tweet" about. Since then, however, I've come to use it more as a quick way to stay informed. I follow a number of interesting people who frequently update their accounts. There are publishers, newspapers, academics, politicians, and even celebrities who tweet. I'm not sure that anyone following me gets all that much enlightenment, but I certainly get a lot out of following others.
Here's a recent TED talk from Evan Williams, the CEO of Twitter, discussing some of the many interesting ways that people use the service.