There are two possible ways a player can win a hand, he either has the best hand at the showdown or he forces all other players to fold their hands in the betting rounds. His probability to win a hand increases as the probability increases that all opponents fold their hands.
Since the percent share of the pot that belongs to a player based on the strength of his cards is called pot equity, the percent that belongs to him based on how likely his opponents are to fold, is called fold equity.
When calculating the expected value, one would reckon on winning a pay-off of x with a probability of, say, 10%.
Two players are in the final betting round before the showdown. Player A has a 20% chance of winning the showdown. Player B will fold 10% of the time if A bets, otherwise he will call. The pot is $4, the bet size is $1.
EV(Bet) = 20% * Payoff(B calls and A wins the showdown) + 10% * Payoff(B folds)+ 70% * Payoff(B calls and A loses showdown)
EV(Bet) = 20% * 5$ + 10% * 4$ + 70% * (-1$)
Hence, the player will win $5 20% of the time, $4 10% of the time, and lose $1 70% of the time. All in all, he will win 70 cents on a bet of one dollar.
Compared this with the (constructed) case where B neither bets nor raises and A just checks:
EV(Check) = 20% * 4$ + 80% * 0$
EV(Check) = 0.8$
EV(Check) > EV(Bet)
So in this example, it's better for A just to check.