Sure, the old Twitter had a bunch of nonsense, but in the last year or two, there are an increasing number of "tweeters" who tweet useful information, in particular links to interesting blog posts or news stories. And since you can "follow" only who you choose, Twitter can be used as a great source of news, but with only the news you care about. For example, a lot of people (especially men) only read the sports section of the newspaper. With Twitter, these people could follow various sport columnists, athletes, and sports media outlets, getting the news that they care about in near-real time. This is how I use Twitter: I follow a bunch of technology, geography, and science folks/businesses, and I often know about breaking news in these fields a day or more before the masses. Plus, I get valuable information in one spot without having to poke around dozens of different websites looking for the articles that interest me. The 140 character limit really helps make for fast, concise reading.
So how should you use Twitter? I advise against using it to communicate with friends and to say what you are doing. For the average person, that is what Facebook is for (and where most of your friends are anyway, rather than on Twitter). Instead, unless you are a business with customers, just use Twitter as a way of keeping track of the stuff you care about. Another great function of Twitter is the ability to search everyone's tweets (not just people you follow or people who follow you). This is a great way to "plug-in" to a significant current event, getting the latest information before nearly everyone else, and perhaps contributing information of your own. Doing twitter searches by location to "plug-in" is an incredible way to gain valuable real-time information during significant weather events, such as winter storms, hurricanes, or tornado outbreaks. You can also search twitter to see what people might be saying about a topic of interest to you. For example, you're wondering if it's worth going to see the new Harry Potter movie; search for tweets related to the movie to get opinions.
In summary, don't dismiss Twitter as a bunch of banter about nothing like I did. Use it to keep in touch with the latest in subjects you care about.