This short format is what attracts people to twitter over using IM's or email messages. Plus you can easily find people who share similar interests and "follow" them to receive information on your topic of preference. Because you are not emailing or or instant messaging, this allows you to share information with people you would have not ordinarily been in contact with. This makes information easier to get a hold of without the restrictions of only those on your email or IM contact list.
The messages you broadcast can be shown directly on the Twitter site itself, as a single SMS message(via cell phone or instant message), or third party applications like Snitter, Twhirl, or the Firefox add-on Twitterfox. You can also restrict messages to be only shown to your circle of friends, but by default its open access.
The tweets you broadcast are shown on your profile page, on your followers pages, and the Twitter public timeline.
For more information on what Twitter is about check out the following sites:
Twitter on Wikipedia
How Twitter Was Born
Official Twitter Blog
Twitter Broadcasting Tools:
TwitterFox Firefox Extension