A gameserver, is the server that the game you are playing on is. Gameservers can either be client run, or rented from many different company's. Ex: Griffenrun, Nuclearfallout (personal favorite) etc.
Simply put its a server or place to play for an online game. A good example would be Counter Strike. Every time you play your playing on a server.
yea , on some games u can host it your own , in most cases you have a serious laggy server if you host it your own , thats why most people buy it
what they say above, i think every fps (first person shooter) is hosted by a person not the company that made it, mmrogp is diffrent though you might only have 4 servers that the company that made the game hosts, thats why you have to pay monthly.