I think of Twitter more as a combination of instant messaging and IRC chatting. It's like being able to send an IM to a few hundred people at once.
If you're involved in blogging or podcasting, and doing it seriously, it's also a great way to network with "like-minded people," but that wasn't really the original intent. It happened that way because the people who are "in" in new media and technology were early adopters of Twitter (back at South by Southwest in 2006, IIRC).
I have
my Twitter account linked to all of my blogs (my
main blog which hasn't been updated in way too long, my
reviews blog, and my
Blogcritics stuff) and my podcast, so that every time I post something new, it sends out a tweet. Just the title of the post and a link, nothing annoying, but it lets the people who follow me know that I posted something. Other than that, it's a lot like SJ said -- whenever you think of something funny, or witty, or something like that, you can tweet it.
There's also the interaction with other people -- people you'd never meet in real life (unless you're Sly). At least one book has been written because of Twitter (and almost entirely written via Twitter, at that). AND I know of a few people in technology who have gotten job offers because of their Twitter account.
Really, Twitter can be whatever you want to make of it. I just marvel at the thought that a company that has never made a dime in actual revenue can keep attracting venture capital; it reminds me of the 90s.