The auction can really heat up, but then comes the play! The side that outbids the other splits into a declarer (who bats for that side during the play, playing her cards, and her partner's!) and a dummy (who basically takes a rest, gets drinks, and exposes her cards for all to see and her partner to play). The partnership who was outbid plays as separate defenders, keeping their hands to themselves. So the play involves two defenders and a declarer all staring at the dummy's hand (called, amazingly, the dummy, coined by some anonymous groundbreaker), and at the cards played to each trick.
If the declarer fulfills the contract, the level determined by the auction (SOLD!), her partnership receives points. The higher the contract, the more points they get. Overtricks, or surplus tricks taken above and beyond the number contracted, are worth a little but not much. Further, a partial, a small contract of about 7 to 9 tricks, is worth a base amount, like minimum wage. The big payoff is contracting for a game, about 10 or 11 of the 13 tricks, or even a slam, 12 or all of the 13 tricks. The defenders, of course, are rewarded for aggressive bidding and/or good play by collecting points for setting the declarer when she doesn't make her bid. Defenders with attitude (or high cards) can even double the declarer, if they really think the contract is not making. Doubling raises the stakes, but a redouble by one of the doubled side re-raises the stakes, effectively a snub of the nose at the doubling side. There is no "re-redouble"; attrition has its limits.
There are three main methods of scoring. First, rubber bridge (don't laugh) is where partials can add up to games, so you can build a game through a couple small contracts, not all in one go. There are no direct game bonuses, but the first side to win two games gets the rubber bonus. To add injury to insult, winning a game means both sides have to start a fresh game, and the potential of the leg, or partial scores collected toward game, of the other partnership is erased. Once one side has won a game, they are vulnerable, reflecting that their penalties are more dear for the rest of the rubber. Of course the side with no games is not vulnerable (more Nobel Prize creativity) until becoming vulnerable by winning a game.
Rubber bridge was the original scoring method, but then people came to think, "Why is winning depending on getting all the high cards?" Thus was born duplicate bridge. Many decks are shuffled and dealt into hands, but then fixed in little carriers (boards) so that many different partnerships can remove a set of four, play them (carefully retaining the cards), and replace them to be replayed later by others. This method allows comparison between scores for the deal, so skill measures into it. Vulnerability, which dictates rewards and penalties, and dealer rotates through a set rotation, and partials do not add up to games, like in rubber bridge. Suffice to say, duplicate is a different style than rubber bridge, and competitive bridge is determined in the duplicate, rather than rubber, style.
To complete the list of scoring methods, there are two others along with rubber, and they are based on the duplicate style of play just described. In matchpoints, which is most common in clubs, pairs play against other pairs at many tables in a determined pattern, and a list of scores for each board is compiled as each pair plays that board. The scores for each direction (North-South or East-West, by convention, the boards being clearly marked so as to keep everything straight) are compared, and ranked from best to worst. If 10 pairs played a board, the top pair has beaten 9 other pairs and gets 9 matchpoints. Another pair may have beaten 4 pairs, and so gets 4 matchpoints. The pair that gets the most matchpoints in the end has won. Overtricks, seemingly trivial at rubber bridge, are huge in matchpoints, since squeezing out one extra trick can mean taking all the matchpoints on the board.
The final method of scoring is International Match Points (IMPs). The scheme is designed for two teams of two pairs each to face eachother. North-South at one table is on the team with East-West at the other table. Deals are played at both tables in turn, and the difference in scores at each table are converted into IMPs by a scale of ranges (0-10 = 0 IMP's, 20-40 = 1 IMP, etc.). The side that plays a board better wins IMPs. A misplayed game is a huge difference, but overtricks are still chicken feed. IMPs is more similar to rubber bridge than matchpoints in this respect, but it still lies in the duplicate family of scoring.
Details on the rules of bridge may be found in your local library, or on the internet at various web sites. I will find some good sites and post them, so check back soon!