One sentence answer: You post a short message on twitter, your friends/followers get a real time instant update of what you've written.
A little more verbosely: Twitter lets you post short messages to its web service. The messages can say anything, but are limited to 140 characters only. Typically these are updates on where you are, what you're doing, whatever. They can also contain links to things on the web like, for instance, the picture you just took of a huge duck sitting on top of the Tel Aviv Municipality building.
You can post these messages directly to the twitter website, or you can do so by sending an email or an instant message or an SMS message (!). There's also a bunch of cute apps you can install on your desktop to do this. Posting is quick, easy and mobile.
Anyone on twitter who's chosen to follow you can get your posts instantly – either in their twitter page, by email, by IM or by SMS. Typically these are your friends and your friends' friends and so forth.
A typical example of using twitter might be that you're going to a bar with a friend. You update on twitter and a bunch of other friends who are near by now decide to join. Then you get an update about a free party near by and y'all head on over there.
Geeks like me can replace "bar" with ".net user group meeting" and "party" with "Sergey Brin talking at GarageGeeks".
Actually, twitter is more about conversation. You send your message out to the world (or @someone specific) and the content is read by anyone following you and they can join in on the conversation. Great fun.
I've been having a lot of fun following and being followed on twitter. It takes a short while to get the gist of it, but I heartily recommend it for everyone. See you there