There is one core use-case for Twitter which I have discussed before, that of using Twitter as a "social network maintenance" tool, to engage in an ongoing conversation with friends and colleagues, across distance and time (some of the people I tweet with are in England, others in California).
1) Real-time news: if something happens, you should be able to pick out reports about it (with the right keywords) by using search.twitter.com
2) Similarly, through the use of hashtags, you can get reports from events and about topics.
For example, if you want to know what people are saying at the UK Serials Group conference, a search of #uksg09 will help (I don't know what you will find there, as at the time of this writing, the March 30 event hasn't started yet - right now you just see people doing their final arrangements to get there).
You can also see an example of this in my previous posting, about the Twittering of Obama's first visit to Canada.
And it was by watching event reporting on the #ali social media conference in Ottawa on Twitter that I got connected into the community that is now working on #changecampottawa. In fact, a lot of the ChangeCamp Ottawa planning and information sharing takes place through Twitter.
Likewise, if you're interested in a particular topic, say #nasa space programs, a search will lead you to other interested people and links about that topic.
You can see a small list of Ottawa social media tags in my posting connecting to Ottawa social media community through Twitter.
3) Mobile and the constraints it brings
A big deal is made out of "only 140 characters". Well, keep in mind that it's possible the desktop computer may one day be considered as obsolete as the deskset telephone. For many people, their entire telephone experience is mobile, and it may be that our main computer interface in the US and Canada becomes mobile as well, as it already is in other places in the world. Mobile means two things: small screens and terrible keyboards. As anyone who is unskilled in cellphone text entry can attest, 140 characters starts to look like a lot when you're trying to use a numeric keyboard to enter words. As well, trying to read more than a sentence or two at a time on many cellphone screens is simply not possible.
The reason both Twitter and Facebook have short status messages, is that this supports SMS as transport, and that many of the users only ever see those messages through the tiny window of a cellphone screen; they never use big screens on desktop computers for their social networking.
4) Other interfaces
The Twitter web interface is pretty basic. There are better clients on mobile phones and on desktops. In particular, TweetDeck is one popular power-user Twitter interface. Here's a selected view showing some live searches (you can click through to Flickr then select All Sizes to see it at original size):
Using Twitter in this way is really a qualitatively different experience. Not only is it much easier to do some things (such as retweet), you also can get a big picture of what is going on - incoming filtered tweets next to your incoming @ replies, next to direct messages, with the option to have people sorted into columns by group (work, personal, etc.) as well as to be continuously running realtime searches on topics of interest. This is a very powerful window on the world of information, if you want to be grand about it, it's a sort of realtime world dashboard.
UPDATE: 5) Search and the Twitter ecosystem
I should mention a couple things:
First about search -
* Twitter search is (mostly) an exact match. If you search e.g. social, you won't see hits for socialize.
* If you use Twitter search from the home page (or add &source=serp to the search URL), Twitter will show matching user accounts as well, and those are sliding matches. e.g. here is a search on #gov that shows matching accounts for various US Governors.
Secondly, because Twitter was smart enough to provide an API, a huge ecosystem has grown up around Twitter - this includes tools I have mentioned before, but many many more. See e.g. this giant list.