In bridge, the bidding often designates a suit as the trump suit. If the final contract has a suit associated with it, spade 4, heart 3, diamonds 2, or even clubs 1, for example, that suit becomes the trump suit for the entire hand.
When a suit becomes the trump suit, any card in that trump suit potentially has special powers; any card in the trump suit can win a trick over any card of another suit. For example, suppose that spades is the trump suit and a player with the Ace of hearts. You can still win the trick with the 2 of spades (assuming that you have no hearts in your hand and therefore can't follow suit).
Because trump suits have so much power, naturally everyone at the table wants to have a say in determining which suit is declared the trump suit. Because bridge is a partnership game, your partnership determines which suit is the best trump suit for your side.
When trumping can save the day
You can easily see the advantage of playing with a trump suit. For example, if you play a hand at a notrump contract, the highest card in the suit led always takes the trick. If an opponent with the lead has a suit headed by all winning cards, that opponent can wind up killing you. She can just keep playing all her winning cards — be it four, five, six, or seven — taking one trick after another as you watch helplessly. Such is the beauty and the horror of playing a hand at notrump. You see the beauty when your side is peeling off the tricks; you experience the horror when your opponents take trick after trick.
However, when the bidding designates a trump suit, you may well be in a position to neutralize your opponents' long, strong suits quite easily. After either you or your partner is void (has no cards left) in the suit that your opponents lead, you can just play any of your cards in the trump suit and take the trick. This little maneuver is called trumping your opponents' trick (which your opponents really hate).
Even if you have a great hand in clubs, if you play a hand in notrump, there's a chance that your opponents can get the lead and race off winning tricks in a suit where you're weak in both hands, such as hearts.
You and your partner need to communicate accurately in the bidding to discover which suit is woefully weak in both hands. When you both are weak in the same suit, you need to end the bidding in a trump suit so you can stop the bleeding by eventually trumping if the opponents lead your weak suit.
When trumping can ruin your day
Bear in mind that your opponents can also use their trump cards effectively; if they hold no cards in the suit that you or your partner lead, they can trump one of your tricks. Misery.
After you have the lead, you want to prevent your opponents from trumping your winning tricks. You don't want your opponents to exercise the same strategy on you that you used on them! You need to get rid of their trumps before they can hurt you by playing your high trump cards early. This is called drawing trumps.