There are many things that make Twitter special, but the main ones for me are:
1) The asynchronous nature of Twitter networks models real life quite well - you might be interested in everything a person has to say, but they don't even know/care about you. This works with Twitter, because following is a one-way concept.
2) This makes it hard to get spammers onto Twitter, because the only way for spammers to get their posts listened to is by having people follow them. Also, services like the Twitter Black list exist to help people sort the wheat from the chaff.
3) The best "feature" of Twitter is its tabula rasa attitude to what you post - as long as it fits in 140 characters, Twitter doesn't care. This has allowed people to build fun minigames (@lyricbot), query/response info bots (@weatherbot), and plug Twitter into their external apps (Remember the Milk, Buxfer, etc). This sets Twitter apart from something like Pownce, which had a similar idea (complete with the asynchronous fan/friend model). The difference (IMO) was that Pownce tried to impose some sort of order on its users, which limited the way it could be used.