A very good and detailed explanation of the game rules can be found here: http://en.wikipedia.org/wiki/Yahtzee.
Please find here a short extract (source Wikipedia):
On each turn, a player gets up to three rolls of the dice. He or she can save any dice that are wanted to complete a combination and then reroll the other dice. After the third roll, the player must find a place to put the score (though he or she can choose to end the turn and score after one or two rolls, if desired). If the resulting combination of dice will not fit in any unused scoring category, the player must place a "zero" in one of the unused boxes. Each player's total score is calculated by summing all thirteen score boxes.
Basic rules
The Yahtzee scorecard contains thirteen boxes divided between two sections: the upper section, and the lower section.
Upper section
In the upper section, each box is scored by summing the total number of dice faces matching that box. For example, if a player were to roll three "twos," the score would be recorded as 6 in the twos box. If a player scores a total of at least 63 points in these boxes, a bonus of 35 points is added to the upper section score. Though 63 points corresponds to threeofakind for each of the six die faces, a common way to get the bonus is rolling four (or five, often using a "Yahtzee as a joker") of a larger number so that fewer of the smaller numbers are needed (a player can earn the bonus even if a he or she scores a "0" in an upper section box).
Lower section
The lower section contains a number of pokerthemed combinations with specific point values:
Category

Description

Score

Example

ThreeOfAKind

At least three dice showing the same face

Sum of all dice


FourOfAKind

At least four dice showing the same face

Sum of all dice


Full House

A threeofakind and a pair

25


Small Straight

Four sequential dice (1234, 2345, or 3456)

30


Large Straight

Five sequential dice (12345 or 23456)

40


Yahtzee

All five dice showing the same face

50 First Yahtzee only


Chance

Any combination often acts as discard box for a turn that will not fit in another category (thus the name), although during a lucky game it can be used to record a high score

Sum of all dice


A certain combination can be scored in more than one appropriate category; e.g., a full house can be scored in the Full House, ThreeOfAKind, or Chance categories.
Upper Section Bonus
If a player scores a total of at least 63 points in the Upper Section, a bonus of 35 points is added to the upper section score. Though 63 points corresponds to threeofakind for each of the six die faces, a common way to get the bonus is rolling four (or five, often using a "Yahtzee as a joker") of a larger number so that fewer of the smaller numbers are needed (a player can earn the bonus even if a he or she scores a "0" in an upper section box).
Extra Yahtzee Bonus
If a player scores a Yahtzee and places it into the Yahtzee box, each additional Yahtzee is awarded with 100 points during the rest of the game. In this case for example the combination 2,2,2,2,2 used in the box Ones counts as 100 points. The extra Yahtzee bonus amends the values of the boxes. Similar to the Upper Score bonus, the probabilistic value of a box depends on the already obtained upper section score.
