A player is said to cover another player if they have more chips in play at the table than the other player. Likewise, a bet is said to cover another bet if it is larger than the other bet, or larger than the other bet could be.
A synonym for cover would be match, when used in the context of bets. It can be used almost anywhere the word "match" would be, as in "I can cover only half of your bet, but I'll go all-in" or "You can have as much of the action as you can cover".
The term cover is generally only used in the context of an all-in bet, where one player is trying to determine what would happen if they called another player's all-in bet and lost: if the other player has this player covered, then this player would lose all their money or chips. If this player has the other one covered, the reverse would be true. This is a particularly important concept in tournament play where losing all your chips means you bust out of the tournament.