[This is a rare public entry.]
Many people I know seem to be curious about what exactly Twitter is, and why they might want to use it. That's not easy to do; I myself was skeptical of the medium (I tend to resist popular trends, at least temporarily), until I gave in and tried it.
I will, however, give explaining it a try.
In short: Twitter is a combination of instant messaging, blogging, social networking, and text messaging.
Twitter allows a person to create a blog of sorts; however, the entries - called tweets - are not of arbitrary length (like, say, on LiveJournal), but have a fixed maximum of 140* characters. This limitation allows Twitter to be posted to - and read (an action called "following") - on almost any device, ranging from a full-size PC, to a simple cell phone via text messaging (SMS).
The social networking aspect comes in because each user has a "friends list" of sorts, called a "following list." Like LiveJournal's friends page, Twitter gathers the tweets of all the people a person follows into a single list, called a timeline. When reading from a PC or most "smart" mobile devices, the timeline is displayed in list form; however, when a cell with SMS is being used, each tweet from a followed person is sent as an SMS message.
When a person tweets - which can be done via many methods, including sending an SMS message to a special number via cell (which is free of charge, as long as you already have an unlimited texting plan on your cell) - it is put in the gathered timeline of your followers, and dispatched by SMS to those of your followers utilizing Twitter via SMS.
The similarity to instant messaging - where Twitter really becomes interesting, in my opinion - is because there are two main methods of directly interacting with other Twitter users.
You may notice that my LoudTwitter reposts to LiveJournal (if you're on my LiveJournal friends list) have a lot of tweets starting with @. This is the first form of direct interaction - called a reply, directed tweet, pointing a tweet, among others. Simply put, having @[username] in a tweet will cause it to show up in a special section on [username]'s Twitter.
(Traditionally, the @ portion is placed at the beginning of a tweet; however, it will be detected anywhere in the tweet. It's also possible to use multiple @[username] terms, and point the tweet at multiple users.)
There is one important thing to remember about @ tweets, though - they are public! If one wants to speak privately via Twitter, there is a way: Assuming the person you want to speak privately to is following you (this is a requirement!), you can send what's called a Direct Message, or DM. These don't show up in any public timeline or log.
To conclude, Twitter puts the best developments in the past decade of interactive, user-driven communication into a system that one can use anywhere. It's pure brilliance.
*: The 140 character limit applies to languages that use the Latin alphabet, like English, Spanish, French, Italian, Portuguese, etc; I don't know for sure how Twitter treats other alphabets or character sets. Japanese ideograms seem to be supported; traditionally, SMS in Japanese allows only half the normal character limit of the Latin alphabet, because each character requires two bytes (instead of Latin's one byte) to express it electronically.