Most games have no dominant strategy. What is best for you genuinely depends on what the others do, and reasoning stalls in a loop: my move depends on yours, yours on mine. John Nash cut through it with a definition that is now the central concept of the subject.
Stability, not optimality
A Nash equilibrium is a combination of strategies, one per player, such that no player can improve their own payoff by changing their strategy alone, with everyone else's held fixed.
The test is local and individual. Stand in the outcome, let each player in turn ask "could I do better by switching, given what everyone else is doing?", and if the answer is no every time, the outcome is an equilibrium. It is a resting point: no single player has a private reason to move.
Applying the test
In the prisoner's dilemma, both defecting is a Nash equilibrium. Neither prisoner can improve by unilaterally staying silent, since that only lengthens their own sentence. Both staying silent is not an equilibrium, even though it is better for both, because either can improve by defecting.
That gap is the crucial point. An equilibrium is stable, not good. A situation can be locked in place by individual incentives while everybody in it would prefer to be somewhere else. Nash equilibrium is a prediction about where a situation settles, not an endorsement of where it settles.
Why it matters
Nash proved in that every finite game has at least one equilibrium, once you allow the randomised strategies of the next lesson. That guarantee is what makes the concept usable: whatever game you write down, the tool will not come back empty-handed.
Its reach is enormous. Auction houses use it to design formats that discourage gaming. Regulators use it to predict how firms respond to a rule rather than assuming they comply. Biologists use it to explain why a population settles at a particular mix of aggressive and cautious behaviour, with natural selection playing the role of the rational player.
More than one resting point
Games often have several equilibria. Two drivers approaching on a narrow road can both keep left or both keep right; either is stable, and neither is better. Then the question shifts from what is rational to what is expected, and answering it needs convention, communication, or a shared sense of the obvious. Which equilibrium a society lands on is often a matter of history rather than logic.