PlayingCards/Bridge

I will not describe specifics of scoring, since I do not remember them. Perhaps someone else could oblige..? Packs of Waddington's Playing Cards usually include a bridge scoring card listing scores and penalties for all the bids.

General

The object of the game is for your team to attempt to win the most tricks possible, having first stated a guess as to how many you are going to win; or to attempt to prevent the other team from winning the amount of tricks they said they could. Failure to win at least the number of tricks guessed brings on penalties; guessing close to the number of tricks actually won is far better rewarded than a spectacular performance after a conservative guess.

Ace is high. Trumps, if any, are strictly higher than nontrump cards.

No communication other than the placing of bids, and requests for dummy's moves, is permitted during the game. In informal games, conversation can be on any topic except cards :)

Starting the game

Four players divide into two teams of two. The entire deck is dealt out. The player to the left of the dealer opens the bidding.

Bidding

Each player may, in turn, place a bid or pass. Each bid must be strictly higher than the previous bid. A bid is a combination of a number and either a suit or "no trumps". The number signifies the number of tricks the player believes their team can win, and the suit is the suit they would need to be trumps in order to obtain that number of tricks. To complicate matters, teams may only bid for the majority of all tricks, and the number used to bid is actually 6 minus the number of tricks they think they can win. So a bid of "1 hearts" means "I think I can get 7 tricks if the trumps are hearts".

Bidding for a higher number of tricks than the previous bid is always permitted. Bids for the same number of tricks are ordered, lowest to highest, as follows:
Clubs
Diamonds
Hearts
Spades
No Trumps

The lowest possible bid, therefore, is "1 Clubs", and the highest is "7 No Trumps".

As soon as four passes in a row occur, the bidding ends and the game begins. The player that placed the winning bid plays first. The other player on his/her team reveals their hand to all players, placing it face up on the table; it is customary to sort the cards by suit. At this point, they are the "dummy" - they make no further decisions that game. The player that won the bidding states which card they wish the dummy to play each time it is the dummy's turn. In informal games, the dummy usually goes and peeks at their teammate's hand about now.

Game

With the exceptions described above, the game is standard /Whist. Players retain the cards they win in order to count them later.

Scoring

Rubber:
The scorecard has a line on it, dividing it into two areas, which are referred to as "above the line" and "below the line". Simply put, points for tricks bid and made go below the line, and others (undertricks by the opponents, and bonuses) go above. If you're doubled, the doubled points for tricks bid go below.
100 below the line constitutes a game. You now rule a new line underneath the bottom, resetting the score below the line to zero for both partnerships. Two games makes rubber, and resets the games score. If you have a game in an uncompleted rubber, you are vulnerable, which affects certain scores.
There are bonuses for rubbers, for games from uncompleted rubbers, and for points from uncompleted games when you call it a day. At this point, after the bonuses are put above the line, you add up the points above and below (included completed games and rubbers) to find the winner.
Score card available at [1].

Duplicate:
The idea in duplicate is that you keep your own hand, and the hands and players rotate so that each partnership plays each hand once; the vulnerability of the partnerships is a property of the hand, and not of your performance so far. Each hand is scored (no above and below the line), and then the NS and the EW partnerships in the hand are ranked and allocated points by ordering. Thus what matters isn't the absolute score on the hand but the score relative to the other people who had the same cards. These ranking points are added over all the boards to determine the winning partnership(s). (Some rotation systems have a partnership being NS or EW consistently, in which case there is a winning NS and a winning EW partnership). If playing at a club, you'll probably have bid cards which have the scores on the back (scores for missed contracts are on the back of Pass). Otherwise, there's a full table at [2].