Looks like Twitter is now becoming quite popular in Europe too. At first sight it seems like a waste of time for most people, but only if you take some time to check it out will you really know if that is true or not.
Here a list of the things you can do to get your feet wet:
Introductions: read the Twitterhandbook first, then you can also go through the new Twitter 101 for business. Also great tutorials can be found on Slideshare, which I recommend reading after these two introductions and after you have actually signed on and started using it already. There are more than 4,000 presentations on Twitter already uploaded. (a nice one is this here.)
Using your first apps: some apps you might want to install are Twitpic, which lets you upload pictures and also send them simply via e-mail. To start getting some discussion going you can try out TwttrStrm, which lets you post questions. You can also link to your Facebook account with twitter/badges. There are endless apps available. One recent one is Geo Chirp, which allows you to search for Twitter users locally.
Install on your mobile: to make things really neat you have to install a Twitter client on your mobile. I have tried a few apps on my iPod touch and all of them do the job quite well. It is more a matter of personal taste I think what app you pick: I use mostly TweetDeck now but Tweetie and Twitterrific are good choices too.
Only time will tell if Twitter is indeed the most initially undervalued business app or the most hyped up tool.
My feeling is that either Google or Microsoft might acquire them if not even Facebook, who seem to have some close links to them already.