Well, the alpha version of the client was just completed last night. A few lucky people are helping me to test it. I'd like to get a few more to help out with it. At this stage, if you get in early your opinions will really help me shape the final product.
Twitter got started on the idea that many people were starting to communicate to their network of friends via status messages. Ever guilty of changing your IM screen-name or away message to let people know what you're up to? That's sort of the idea behind it and maybe you'll feel right at home using twitter. It lets you keep your network of people updated on what you're doing.
Twitter is many things to many people. For me, I don't post very often in a blog because I feel obligated to say something. If I don't have anything interesting to write, I feel like I'm just dumping blogarrhea all over the net. It makes me feel like a gimp to just post a "Going out for a hair cut after work and will probably watch the new episode of House tonight" kind of post. Twitter changes that -- it maxes at 140 chars and is meant for those little updates. I can keep people in the loop and not futz up my blog content.
Twitter is being used for a lot of things. Like flash-mob style coverage of events. I made an SL client because I thought it might be a cool way for people to keep their avatar friends up to speed on what they're doing. I'm sure there are many other applications like perhaps a SL-Maps mashup in the future (like this one which appears to be missing at the moment).
Other cool uses of twitter here and here.