The short answer is that it's a microblogging web site where anyone can post really brief (up to 140 characters) blog entries, called "tweets."
It's easiest to follow if you are using Twitterrific or a similar tool on your computer that shows you new tweets shortly after people post them, so you're just always kept current on what's going on. (You can also just follow on the web, periodically checking your home page as you'd periodically check your LJ friends page.) As with LiveJournal, you can have a "friends list" (a list of people you "follow") and you can decide whether your posts are public, or restricted only to the people you follow.
Since I say a few things there that I never get around to also saying here, I have an automatic post set up every night that will, once a day, post that day's tweets here. I do that because some of them are things that seem worth sharing. The fact that several of them have sparked comment threads on LiveJournal convinces me that this is worthwhile, despite the confusion and despite the fact that at least one of my LJ friends finds Twitter itself incredibly distasteful for some reason I haven't figured out. If you don't find these little postlets interesting, you're welcome to ignore them.
Most of it's more interesting if you follow it live throughout the day, but catching up nightly seems like it should be adequate for most people who don't already use Twitter. If you do use Twitter, feel free to follow me directly at http://twitter.com/mhaithaca