It's a computer, usually in a place with good connections and fast internet, that is used solely for the purpose of running a specific server. A lot of companies have dedicated servers to get the ball rolling for their multiplayer games, but that's not what the controversy is over. Anyone can set up (or, if you live in less than ideal conditions, rent out) a dedicated server, then they have complete control over it. It's wanted so much because it is the key part of clan gaming and modding communities, as well as providing a better play experience to players. Player hosted dedicated servers have also kept a few games I know running long after the developer has apparently abandoned it (meaning only company owned dedicated servers ala Bad Company 2 is bad, though DICE seems to support their games for a pretty good amount of time, making it less of an issue).