It's a public messaging system. The best use of it (for people who don't want to tweet) is for keeping up with news, especially in categories you want to follow. For instance, I follow a lot of news publications and tech writers. If you are into sports, you can follow sports beat bloggers and journalists. Or practically any other community or niche.
You can also follow celebrities, politicians, activists, etc. of course, which is really what Twitter is known for - that, and for having almost as many spammers and trolls as YouTube (or maybe more) - for good or for ill.
And, like John Connor mentioned, it's a good potential channel for support. A lot of companies and organizations are also more active via Twitter feeds than even their own sites, too, so it's a really good way to find out about service outages and scheduled maintenance or other issues in real-time.