As the name Tweetname suggests, it is a service that offers domain name registration via Twitter. All you need is an account on their site and send a direct message with the name you want to the @tweetname Twitter account. You will get an instant reply whether the domain registration was successful or the name is already taken.
With Tweetname domain registration has never been more ubiquitous having hundreds of Twitter clients around. Whether you want to rely on Twitter for a domain registration is another question though.