Trick-taking games are generally played with (some or all of) a standard deck of playing cards. For one "trick" each player plays a single card, following suit if able, and the highest ranked card wins, taking the trick and keeping it facedown until the end of the round.
Some examples of trick-taking games are Hearts, Bridge, Euchre, Spades, and Sheepshead. Each game may have a particular variation on the basic trick-taking rules, such as the inclusion of a trump suit (a suit in which all cards are ranked above other suits), and the play options if a player does not have any cards of the suit intially played. Wikipedia has a nice article on the subject.