Twitter is a micro-blogging application, which is basically an easy and quick way to send out an update or share something, similar to what you would do on a blog but much more lightweight. Though it does come without all the weight of a full social network, Twitter offers things like SMS updates, IM updates and a range of applications built on their API, so you have many ways to interact and update. You can also send direct messages to users and easily reference/reply to others by using @username (such as "@akamike: great blog post!").
An example use would be when you have acquaintances or "fans" whom you like to keep up-to-date on various things, to embrace them in some way without giving them full access to your life (i.e. Facebook). You could have them all follow your Twitter account and keep them up-to-date from there.
Another would be for stubborn friends who won't "conform" to the social network of your choosing. You all still need a way to share status updates and links/cool stuff. You could all follow each other on Twitter and just send out a message to them all (free, excluding SMS) quickly and easily. You may argue that you can just do that with SMS, but as I said that isn't necessarily free like Twitter and you don't really want to send an SMS to all your friends saying "lol im playing wow ^^".
It could also be used by businesses as a way to keep fans/users up-to-date with products and happenings with said business.
BBC News is aggregated on a Twitter account, for example, though it's an unofficial account.
In my blog post on micro-blogging, which I linked to in my earlier post, I analysed a number of services, since Twitter is not the only option, and broke down the features they offer.
Gaming World Forums
Let's share Twitter names