To begin, one card from every month is selected from the deck and placed face down on the table. Each player selects one card; the player who selects the earlier month is called the 'oya' (parent) and will play first.
To begin a hand, each player is dealt eight cards, hidden from view of his opponent. Eight cards are also dealt to the table, face up. These cards are called the 'field', and can be collected by either player.
On a player's turn, he will do two things: play a card, then draw a card and immediately play it. You play a card by placing it on the field. If there's another card in the field in the same month as the card you play, you collect both cards. Place them face up in a pile on your side of the table. If there is no match for the card you play, add it to the cards already in the field. If the played card has three matches on the table, the player collects all four cards for his yaku pile.
The goal in Koi-Koi is to collect certain groups of cards called 'yaku' (sets). Once a player collects a set, he is given the option to end the hand or continue playing by calling 'Koi-Koi'. If he ends the hand, he collects the points his sets are worth; his opponent receives no points, even if he, too, has collected sets. Once a player ends the hand and points are tallied, the cards are shuffled and a new hand is dealt. The player who ended the hand becomes the new oya and plays first.
When a player wants to try and collect more yaku, he calls 'Koi-Koi'. Play then continues as normal until another set is collected. A player can call 'Koi-Koi' as many times as he would like. If a player's sets ever total 7 points or more, the points double. (e.g. A player ending the hand with a group of sets worth 7 points would collect 14 points.) However, if a player calls 'Koi-Koi' and his opponent makes a set before he can make another, the opponent is entitled to double points. (If the opponent's sets total 7 or more points, that would result in quadruple points.) When deciding whether or not to Koi-Koi, a player must balance the additional points he may be able to collect against how close his opponent is to making a set.
Koi-Koi is played with traditional Japanese playing cards known as Hanafuda. It's a 48-card deck, including four cards for each of the twelve months. Generally, each month includes two high-score cards and two basic cards, although there are exceptions. The point values of the specific cards do not directly affect the scoring in Koi-Koi, but they are used to create the sets.
The sets (yaku) in Koi-Koi are [for the most part] derived from the standard point values of the hanafuda cards, although they don't impact the scoring in Koi-Koi. Each set has a given point value, although a few sets are scored based on the number of cards of a certain type collected.