Don
think of it as a "mini blog" -- people write posts of 140 characters or less. A little like the wall on facebook, but without all the facebook overload. It's a LOT less information / toys / obnoxious stuff (oops, showing my bias) than facebook.
You follow whom you want to follow, and you can make your posts private if you only want friends/family to see them.
So if you were to sign up, you make a name/nickname for yourself. Then you go to your home page and it shows you, down below, all the recent updates from whomever you follow, and your own posts (tweets).
You can get "twitter readers" that automatically refresh and show you new posts from those you follow, some categorize them so you can see the important-to-you people in different folders/frames than the ones you don't necessarily want to read each and every post ...
It can be a bit odd, if you see half-conversations that someone you follow has with someone you don't follow. You'd only see the posts from the ones you follow unless you went to read the other persons.
I like it because I can use it like a mini blog and my friends and family read it. When my mom was traveling she used hers to keep the family updated as to where they were. You can send in posts from your cell phone.
Sure, some people post empty, stupid stuff all day long, things like "Just ate a burrito." -- you don't have to follow them if you don't want.
Contrary to what Purdy implied, it doesn't mean that everyone who uses it posts stupid, empty things. Some businesses use it to post links to information. Once again, you follow the ones you're interested in and ignore the rest.
I hope that sort of explains it. You can go to Dr. McDougall's twitter link (posted in the first post) and see what it looks like without signing up (another big benefit over facebook, if you ask me)