Equity in general means your percentage of the chips you would win, on average.
Fold equity is the chips you win the percentage of time that your opponent just folds.
You have zero fold equity whenever you call. You can only have fold equity by making a bet.
Example:
an opponent with pocket 4s raises to 1000. You go all in with AK for a total of 3,500 (or 2,500 more). The blinds fold leaving their 600 in the pot. If he calls AK has 45% equity vs 44. So you'd stand to win 4,100 45% of the time or $1,875. And 55% of the time you'll lose the entire $3,500 you put in so -$1,925 making the play as a whole a loser by -$50. So, on the surface pushing with AK right here is a losing play.
But wait, there's more. As the person pushing all in you have at least SOME fold equity. It's not an exact science, but you have to figure that some % of the time he will fold to your bet. Take a guess. I'd guess that 25% of the time he'll fold.
that means, 25% of the time you'll just outright win the $1,600 in the pot. So your fold equity is $400 (1,600 x 0.25). If you thought he would fold 10% of the time, then your fold equity would be $160. If you thought he'd fold 50% of the time then your fold equity would be $800.
So, you add the fold equity to your regular equity to find out the profitability of the move as a whole. In this example, if we think he'll fold 25% of the time to our shove, then shoving with AK there is $400-$50 for a total profit of $350. Basically, with fold equity on your side a lot of borderline decisions become +EV.
That is why, in general pushing all in is better than calling all in in almost all cases.