The way I use Twitter is fairly standard though I do lock my account to keep it private. It's not for my family, which I reserve for Facebook, just for my friends.
You send messages much like you would on this forum. It's a single message that any and all can see. Then you can reply to a specific person or persons, or just create a thread/message that has nothing to do with any other topic.
You also read news from others. Maybe from friends you personally know and follow or other people and sites you follow jut because you feel they have interesting things to express. Stand-up comedians and writers aree pretty funny with the one liners, but mostly I use it for news and media organizations.
It's much like Facebook's wall and if all you're going to follow are your Facebook friends there is little need. It's much like RSS and Gogole News Feeds and if all you're going to do is get news there is little need. If you want a way to interact with groups of friends and get news in a convenient capsule its great.
A nice feature for celebrities — which helped make it famous and made it hard for many to understand its utility — is that it puts them in control of their own gossip and lets their fan interact with them more directly and considerably faster.
For example, Neil deGrasse Tyson (
@neiltyson). Not only does he post interesting and geeky facts throughout the day, but he also asked questions and replies to his followers.
I won't say that I initially understood Twitter. In fact I thought I was the last person to "get it" but I have been using it for a couple years now I can say with absolute certainty that it has a viable utility (but that doesn't mean for everyone).
One thing I don't think I've seen in this thread is the questioning of one's own feelings toward Twitter. Apple has integrated it within iOS to the point there is a default option for it in Settings. I'd think that would make anyone on this forum who doesn't "get it" reexamine their position.