Basically, the idea here is to figure out your best strategy for the 'game' (which really means any situation where you need to make a decision), given that the other "players" don't cooperate with you in any way (i.e. you assume they are going to act in their own self-interest).
The most basic successful strategy for the Prisoner's Dilemma is called "Tit-for-Tat", in which you always choose the "altruistic" option, unless the other player chose against you last time.
Personally, I plan on pursuing a career where game theory would play an important role in my research or work (teaching or intelligence research).
In many ways it's the science of cause-and-effect (hence why the Tit-for-Tat game solution works so well). By applying the concept of how one event can determine another on a predictable basis, one can help explain and potentially solve many of the world's greatest problems.
guess 2/3 of average: http://en.wikipedia.org/wiki/Guess_2/3_of_the_average
pirate game: http://en.wikipedia.org/wiki/Pirate_game
dollar auction: http://en.wikipedia.org/wiki/Dollar_auction
Most common/well-known:
prisoner dilemma: http://en.wikipedia.org/wiki/Prisoner%27s_dilemma
chicken: http://en.wikipedia.org/wiki/Game_of_chicken
battle of the sexes: http://en.wikipedia.org/wiki/Battle_of_the_sexes_(game_theor...
El Farol bar: http://en.wikipedia.org/wiki/El_Farol_bar_problem