How does a U.S. Senate filibuster actually work?

Lately there’s been a lot of talk about filibusters in the U.S. Senate, but we never hear about any marathon speeches anymore. A filibuster, as I understand it, means you simply continue debate until your opponents tire of it and table the bill and move on to something else. Now it seems that if you don’t have 60 votes, then a filibuster automatically results and your bill fails.

I’ve heard some people say that the Senate leaders should force a real filibuster, make their opponents actually speak for hours on end if they want to stop a bill. Is there a mechanism by which they can do this? Can the Democrats actually make the Republicans (and Lieberman) stand up and read Shakespeare? If they took this approach, what would be required of the filibusterers? Does one person have to do all the talking, or can they pass the torch to others?