A gamer is a person who think that he has spent so much time playing games that he deserves a title: Gamer
The gamer is usually some type of regualar game user with his main focus on competative gaming. A gamer plays to win and like to talk about their top score, lvl, gear etc etc.
Gamers, as has been said, are usually playing online, since singleplayer wont give them satisfaction of saying that they are better than you in a certain game since it does not contain a mulitplayer function.
Places you usually see gamers in are: WoW, LoL, HoN, TF2, BF2, MW, MW:BO and online minigames that has a score board system.
Gamers also buys all kind of "gaming equipment" for there computer, such as : Keyboard with light coming from it and extra set of button, Mouse with light coming from it and extra set of buttons and a large mousepad for 100 $.
The main diffrence of a gamer, and somone that plays games are that gamers are usually self indulgent assholes that play for win and top score, wheras the regular player just plays for fun.
Gamer: Hey johnny, how many HS did you get in your last BF2 game?
Johnny: 2 i think
Gamer: Haha noob, i got 1337, woot lol!