Could anyone explain to me in layman's terms the fundamentals of game theory by John Forbes Nash and others?

Very simply Nash states that given a situation (or game) the individual players will select the options that results in their highest gain regardless of what the other player is doing. Sometimes this leads to the highest gain for both players, and sometimes it leads to a less than ideal gain for both players. You can search for prisoners dilemma if you want to get the basic idea of how a two person game is played and the outcome.