The Twitterverse is simply Twitter. It can be on the computer, your phone, news media, etc. The use and discussions going on on Twitter are often referred to as the Twitterverse (hence the universe). It is everything Twitter related, from the people to the apps to Twitter.com itself.
There is also a website called the Twitterverse.com. As of today, they are under construction. Not sure what they will be doing with that, but it is not affiliated with Twitter. They are just fans, like me. 