1 - Replacement for RSS. Follow the blogs you do on RSS and get push notifications of updates. Authors sometimes have smaller things to say than a blog post requires, so the feed acts as an added feature.
2 - Follow the doings of "niche celebrities" like Eric Reis and PG. They give advice and comment on recent events, but aren't really popular in the mainstream sense.
3 - Follow more general celebrities. It gives people you are a fan of a direct line to you independent of traditional media outlets. Think Conan when he was without a network, and Kevin Smith is doing some interesting things as well. Sure, Kim Kardashian has a huge following which she uses to hock products, but there is a massive audience who wants precisely that.
4 - Comedy feeds. I find when comedians have only 140 characters, the constraint makes the jokes more punchy. (plug: my traction deficient app http://laughlitmus.feedladder.com is trying to capitalize on this; viral loops are a bitch)
5 - Real time events. You'll hear about an earthquake on trending topics before any media outlet.
6 - Following privacy paranoid real life friends. Twitter has a dead simple privacy model. Either people can publicly see your feed, or they have to approve you. Some people I know got fed up with the Facebook privacy shell game and switch to Twitter for this.
7 - Connecting with a loosely coupled community. For those of us not in the valley, finding fellow hackers can be troublesome, and when you do, you don't really need a heavy connection, but following each other on twitter is a nice lightweight way to see if the local tech event is going to be hacker or muggle heavy.
8 - Location based chat for conferences. Hashtags allow for people to mark a discussion about anything, and a lot of meetups will tell you the recommended hashtag so you can see what people are discussing in a specific place.
I hate is the whole "Twitter is just people talking about what they had for breakfast" complaint, it really just meant you are following boring people. There actually exist a non-trivial set of people I know whom I am friends with on Facebook (and real life) but wouldn't follow on twitter because they're tweeting about stuff I have no interest in.
Twitter also hooks you into a social circle of your choice in a unique way. I've heard about all sorts of software projects, new releases, rumors, etc. from Twitter.
I have also used it as a place to post and broadcast messages about my businesses, and customers may see it appear in their feed if they follow us, or they can go to our page as twitter members or not and see the latest news.
As for more intimate discussion, I've never really grokked how to use it in that way.