You can see a map with where people are posting on Twitter from: http://twittervision.com/
You can see a real-time tag cloud showing what topics people are talking about on Twitter: http://www.twitscoop.com
And of course, you can search Twitter to see what people are saying about something: http://search.twitter.com
Case in point: A few weeks ago there was a Yahoo outage on the East-Coast for about 40 minutes. When I first discovered it, I searched Twitter for "Yahoo" and found a dozen people within the past minute complaining Yahoo was down. This quickly confirmed the outage for me.
These are just a few examples, but I think you get the idea. I rebelled against Twitter at first, but now I think it's awesome. The turning point for me was actually a Penny Arcade comic (http://tinyurl.com/6gh4lg) and the follow-up blog post (http://tinyurl.com/5c4zds) that made me realize (http://tinyurl.com/5tj2kk) I don't use Twitter for what they intended (to answer the question, "What are you doing?"). Twitter is exciting because the number of possible things it can be used for are nearly endless and the ways the data can be used are almost equally as endless.
Another awesome usage of Twitter that I've seen recently is the way site admins are using Twitter to leave status updates about the outage. They list their Twitter feed on the "We'll be right back" page, but since it's Twitter, people can choose to "follow" the user and get updates without visiting the site (if they enable "mobile updates", they'll even get notified via SMS). mininova.org is one site I've noticed using Twitter like this.
Companies are finding it useful for guerilla marketting.. which is bad when it's spam but if done right can be a thousand times better than a blog.
You also get to connect with people you never knew existed. Scoble retweeted a comment I made about Oauth and it led ot me having a fun discussion about it (and getting some new ideas etc.) with people who agreed and disagreed. We'll probably never chat again but I found some interesting people to follow and we hammered out ideas really well.
And then it can be just a simple comunications tool. I love seeing what people are up to (in a non freaky way) and Twitter is more real time that a lot of other social networks out there. I think probably due to the # of desktop tools.
Anyway - a month ago I would have said "nah it's a nice idea but full of idiots" now I say: it's still full of idiots but when you get past the spam and self importance (cof john chow cof) then there are some real gems there.
:)
Pretty much a POTs replacement for the Internet with voice mail & conference calls.