Twitter is a service that allows you to tell people what you are doing (or maybe just short notes for yourself, that you don't mind others seeing). Kind of like a mini-blog or status update (messages are limited to 140 characters).
All you need is an internet connection (to set up an account or send updates and check other peoples twitter status) or a mobile phone (to receive SMS alerts as well as update your own status). Updates can be limited to friends only if you don't want everyone on twitter to know and you can also see what others are posting. You can also limit (or even stop) friends' status updates (e.g. when you are sleeping).
If you update your status via your mobile, all you pay is the standard SMS rate. Twitter can also be updated via its website or a third-party application.