Think of it like this:
Dedicated: The whole machine is your's to use, like your computer at home.
VPS: The vps takes a dedicated server and divides it into partitions, kind of like you do when you install 2 OS's on your home computer. If you buy a VPS package, you are buying one partition on a dedicated server
Shared: Basically, this is much like controlling only one folder of your home computer
Now, to answer your question about game servers....
It depends. Each provider is different, and you would need to ask them before purchasing. I can tell you however, that game servers can take up quite a bit of resources, and a lower cost VPS probably will run out of those resources rather quickly if you run a gameserver on it. Most VPS providers do have higher packages however, that should provide enough resources. Again however, it really comes down to if they will allow it, and what your budget is.