(NOTE: There are many different varieties of Rook. To differentiate these rules from those of other varieties, I've coined them "Princeton Rules," for that's where I learned to play the game. Variations on Princeton Rules and other variations are listed below.)
Step One: Beginning
First, the simple and straightforward stuff, for completeness.
The four players should sit around a card table or suitable substitute, partners sitting across from each other. Deal out the first 52 cards, placing the 53rd face down in the center. (Each person should now have 13 cards. Duh.) Arrange your cards in suit order without telling any other player what you have. (Don't worry about what the joker (or "rook") means just yet.) Now we are ready to bid.
Step Two: Bidding
After the cards are dealt, players bid for the right to call trump. Starting with the player to the left of the dealer and going around clockwise, each player either makes a bid, or passes. A player who passes may not re-enter the bidding. This process continues until three players have passed; the fourth is the winner of the bid. The winner gets to call trump (but not just yet! First, I must explain bidding a bit more).
A player's bid signifies the minimum number of points that he believes he and his partner will win in the hand. (More on winning points later.) Bidding starts at 70 and increases by 5 each time someone bids again. There are 200 total points in a hand, broken down as follows:
No other cards are worth points.
So what keeps people from bidding right up to 200 in order to get to call trump? If after playing the hand, the player who called trump and his partner do not make their bid (in other words, they get fewer points than they bid), then they lose the number of points that they bid.
Step Three: After Bidding
After bidding, the winner of the bid takes the card from the center and drops a card from his hand, placing it face down in front of him. This card may be the card from the center. The winner and his partner keeps any points that the dropped card may be worth.
All players must now choose three cards to pass face-down to the opponent on either their left- or right-hand side. The direction of passing alternates between each hand (right, left, right, left, etc.). You may also alternate between passing to the right, to the left, and not passing, as long as the order is regular and agreed upon beforehand.
A player may not pass any cards that were passed to him. The winner of the bid may not look at the cards passed to him until trump has been called. All other players may look at the cards passed to them only after they have passed cards.
After the cards have been passed, the winner of the bid calls trump. Players may quietly curse or cheer within reason. The winner of the bid picks up the cards passed to him, and play commences.
Step Four: Playing
The person who called trump gets to choose whether he goes first or the person to his left goes first. Whoever goes first plays a card (placing it face-up in the center). Each player in turn (clockwise) plays a card until all four have played a card. This is called a trick.
As in most games of this type, all other players must follow suit, if they can. In other words, a player must play a card in the suit that was led if he has any cards in that suit left. If a player does not have a card in the suit that was led, then he may play any card in his hand.
The winner of the trick is the person who played the winning card, as determined by the following rules:
Whoever wins the trick keeps all the cards, and thus all the points in the trick, in a pile kept by him or his partner. He then leads for the next trick. Play continues in this way until all cards have been played, for a total of 13 tricks.
Step Five: Scoring
After play is over, each team counts up the number of points in their pile. A reminder:
So there are 200 points in each game. A running score is kept; usually games go up to 1000 points (first team to reach or exceed 1000 points wins), but any number may be agreed upon beforehand.
If the team of the player who called trump has fewer points than he had originally bid, then they are "set." They do not add the amount of points they get to their total; rather, they subtract the amount of points that they bid. The team that did not call trump has no such restriction.
Example: If you and your partner win the bid at 110 points and you win 110 points in the game, then 110 points are added to your total score and 90 points (the number of points your opponents won) are added to your opponents' total score.
Another example: If you and your partner win the bid at 110 points and you win 105 points in the game, then 110 points (the number of points you bid) are subtracted from your total score and 95 points (the number of points your opponents won) are added to your opponents' total score.
One further scoring bonus: If a team takes all thirteen tricks, then they get a 100 point bonus, giving them 300 points for the game. This does not happen very often, so if it does, one has the right to gloat.
*Information courtesy of Peter