Twitter is a free microblogging service where users publish updates or "tweets" of 140 characters or less, to describe what they're doing. If you "follow" people that you're interested in, you will see their status and you can reply to them. Accounts can be public or private so you can share your thoughts with the world or a small circle of friends.
The short video below explains the basics. The Twitter support site is also a good place to get started.