On Twitter, there is a favourite button that lets you mark something with a star to show that it was a favourite tweet that you really liked/enjoyed/loved. At least that is what people are telling me now.
I use it in a similar way to a bookmark for something I want to read later. If there is something I want to read but can't download because I'm on a phone with 3G or just busy at that moment then I mark it and look at it later.
The best thing about Twitter is that it is whatever you make it. Use it however you want.