One of the things that makes Twitter so instant is that it is moderated reactively rather than proactively. As a result, when you post something it goes into the stream of tweets instantly and can be read by anyone with access to the feed. While this is popular because it allows a steady flow of posts and lets you speak freely, it does have its drawbacks, not least of which is the fact that it does allow people to tweet things that others may not want to read.
Among the negative aspects of this is the fact that if someone takes a dislike to you on Twitter, they can bombard you with unpleasant @replies and encourage their followers to do likewise, or drive a wedge between you and friends. They can be reported to Twitter, and anyone abusing an account will be banned from the site, but this does not preclude them from setting up another account and tweeting from another IP address. Neither does it change the fact that they can do it in the first place.
One solution to this, although it is a partial one and not foolproof, is that you can protect your tweets and prevent people from seeing what you post unless they have been approved by you. It does not immediately prevent people from randomly insulting you, but it does lessen the possibility that you will appear on their radar. Aside from this it is difficult to totally prevent such idiots from harassing you, but remember, they’re just a line of text and if they are devoting their time to attacking you, they’re quite the moron anyway.