NE is a game theory concept, so it applies to games. Economists use the word game a bit differently than other people, so you can think of a game as a scenario where more than one player interact with each other and the payoffs (perhaps who wins and who loses and by how much) are determined by the decisions of all the players. A simple game is played on The Price is Right. The winner is the player who guesses the number closest to the value of the item without going over. All four players make a guess and therefore whether or not one player is a winner depends on the value of the item and the guesses of the other three players. Likewise, paper-rock-scissors is a game. If you throw rock, your payoff depends on what your opponent throws.
In a NE, no player can increase his own payoff by deviating if everyone else keeps their move the same. Think about letting just one player (at a time) have a do-over...if they change their move, then we say they have deviated. Another aspect of NE that hurts brains is that there can be multiple NE in one game. I am going to describe a two player game for simplicity.
You and I get to pick an integer between 1 and 9 inclusive. We do not get to communicate with each other about what we will pick. Say that you pick a number by writing it down and giving the paper to a person organizing the game. The payoffs are as follows: if we pick the same number, then we each get paid that number of dollars (by the person who is organizing the game). If we pick different numbers then we each have to pay the same number of dollars as the number we picked (to the person organizing the game).
Example 1 payoff:
I pick 5 and you pick 7. Now I have to pay 5 and you have to pay 7.
Example 2 payoff:
We both pick 4. Now we both get paid $4.
What about the NE?:
The moves we made in example 1 are not a NE. If I were allowed to deviate, I would change my pick to 7 and we would both get paid $7. If you were allowed to deviate then you'd pick 5 and we'd both get $5. It only takes one of us wanting to deviate to make the scenario a non-NE. The moves we made in example 2 are a NE. If either of us were given the chance to deviate (given that the other player cannot change their move) then neither of us would want to because we'd go from getting $4 to paying something which does not make us better off.
Here's the tricky thing...
First of all, you have to get past the idea that this is a simultaneous move game and you will not know what your opponent has chosen until after you have made your move. Second of all, you might have deduced that any time we both pick the same number in the game above leads to a NE. That is correct. There are 9 NEs for that game. We could settle on any one of them and it would be a NE. All other choices where we pick different numbers are not NEs because one of us would like to deviate if we knew that the other person would not be able to change his answer.
I hope that helps. I can answer any questions in the comments.
Note: I've only discussed pure strategy NE because mixed strategy will blow your mind...and I can't say much about it to the lay person except that it would be bad for a player to commit to a strategy such as "always choose paper in paper-rock-scissors" because the opponent would then crush you by always choosing scissors.