Hole cards refer to the face-down cards given to each player at the start of the hand that only he can see and use.
For example, in Texas Holdem, you receive two hole cards at the beginning of the hand and can use them together with community cards to make the best possible five-card combination. If you receiveQsJs as hole cards and the flop comes Ts 9c 5s, now you have a straight and a flush draw with two overcards, making it a very powerful holding.
For the most part, you only need to show your hole cards in a showdown situation or when you move all in.
The number of hole cards can also vary depending on the format you play. For example, in Pot Limit Omaha, you will receive four cards at the hand's start, but you can use only two of them together with community cards to make the best possible hand.