I wrote a pretty long post at Mises Canada trying to give the average person a real sense of what Nash did in his dissertation. Ron Howard is not the hero of my post. An excerpt:
The central result from the work of vNM was the minimax theorem. The full details are here, but the intuition is: In a finite two-person zero-sum game, there is a value V for the game such that one player can guarantee himself a payoff of at least V while the other player can limit his losses to V. The name comes from the fact that each player thinks, “Given what I do, what will the other guy do to maximize his payoff in response? Now, having computed my opponent’s best-response for every strategy I might pick, I want to pick my own strategy to minimize that value.” Since we are dealing with a zero-sum game, each player does best for himself by minimizing the other guy’s payoff.
This was a pretty neat result. However, even though plenty of games–especially the ones we have in mind with the term “game”–are two-person zero-sum, there are many strategic interactions where this is not the case. This is where John Nash came in. He invented a solution concept that would work for the entire class of non-cooperative games–meaning those with n players and where the game could be negative-sum, zero-sum, or positive-sum. Then he showed the broad conditions under which his equilibrium would exist. (In other words, it would not have been as impressive or useful if Nash had defined an equilibrium concept for these games, if it rarely existed for a particular n-person positive-sum game.)