What is "Nash equilibrium" and what were its impacts on the world?

Game theory is a study of strategies involved in complex games. Almost every human interaction - politics, economics, law and religion  can be modeled as a game. You are in a game, if your fate is impacted by the actions of others. 

There are many ways to classify the games. One such classification is whether the game is cooperative or not. Cooperative games are like partnerships where the players work together and contracts can be used to force a non-cooperative player. 

Non-cooperative games are those where the players enter into no agreement and each one is looking after his/her own interests. Think of Apple vs. Google or US vs. Russia. They are playing a complex game.

What Nash said in his landmark work - Equilibrium points in n-person games - is that even in such complex non-cooperative games there exist an "equlibrium" where no side would benefit by changing its course. At this equilibrium, each sides knows its adversary very well and sticks to its strategy. For instance, during the Cold war, US and USSR were locked into an equilibrium - Mutual assured destruction - where each side knew exactly where the other's positions are but still didn't start a war. 

Suppose my wife and I are trying to coodinate when to reach home. If I go home and she is late, then I waste time. Same if she goes home and I go late, she wastes time. The Nash equilibrium occurs when both of us reach home early enabling us to spend time together or both of us working late making more money. These two circle points are the Nash equilibria. 

[This is called a payoff matrix that shows how many points each player gets if a particular strategy is taken. For instance, if I go home and she goes home early (top-left quadrant) we both get 3 points. On the other hand if I work late and she comes home, I get two points, while she gets only 1 as she is bored.]

By mathematically proving that there exists at least one such equilibrium point in any such game, Nash helped economists, politicians, bureacrats and business strategists understand the world around us in a better way. It formed the basis of many strategies we see around the world. It also indirectly showed how different players can dance to the tune even without the need for contracts.

There can be multiple equilibria in a game. Each equilibrium point is like a lock out of which no player will be able to escape easily. For instance, India and Pakistan can fight each other like cats & dogs to infinity or an equilibrium where they can collaborate in a virtuous loop. When the players are locked into a bad position (such as an arms race) it is called the Pareto inferior equilibrium. 

To sum it up, the simplicity of Nash equilibrium enabled the growth of game theory as a legitimate discipline and enabled strategists & regulators to model their environment better.