As such, I thought I'd round up and recommend a few interesting links.
Firstly, for those wondering what on earth twitter (or the point of twitter) is, here's an excellent cartoon explanation courtesy of CommonCraft.
Tim O'Reilly explains why he likes Twitter.
Pistachio also have an excellent introductory roundup.
TwitTip gives tips on how to use twitter, and who to follow, albeit from a slightly commercial viewpoint.
Next, a few clients I'm rather fond of, as an iPhone / Mac user:
Twittelator Pro currently has pride of place on my iPhone.
On the desktop, Twhirl has recently given way to Tweetdeck - both are excellent, cross-platform and free!
Twit That! isn't exactly a client, but it's a useful bookmarklet that lets you share a URL you're reading in just a couple of clicks. TwitSnip is somewhat smarter, but a bit slower due to remote lookups. And for anyone wanting a full client in the browser, there's TwitFox.
One of the strengths of twitter is its openness (albeit with the auth issues previously mentioned), which means that there's massive potential for websites to integrate with it. Most of these provide ways to track the overall worldview of what's going on on twitter:
Twittervision is a fairly abstract global view with no real sorting, but gives a nice idea of the global reach of twitter.
Twistori gives an "emotional overview" of tweets. Again, abstract but fascinating.
TwitScoop is more structured, gathering worldwide topics in real time to show what's catching the world's attention. It's from this site that I learner about the Mumbai attacks, the arrest of Damian Green, and the recent earthquake off the California coast.
Twitturly does a similar task, but works on the URLs that are being shared via twitter (unpacking them from their shortened forms where appropriate). It's kind of a real-time, automated Digg.
Two more unusual uses are Tweetsgiving (a recent, successful charity intiative) and TweetPay (which appears to be an IOU system).
More whimsically, we have Twitter Image for free and custom Twitter Backgrounds, and Twitter Grader, which is mainly an ego engine.
For more links as I find them, follow my delicious twitter bookmarks.