Simple explanation

Japanese mahjong is a derivate of the Chinese game, and the goal is to be the first out of 4 players to complete a mahjong-hand with those strange tiles. And the idea of the tiles is quite similar to common playing-cards, with the differences that there is only 3 suits that range from 1 to 9.
The three suits are: bamboo, circles and characters. There is also tiles for the different winds: east, south, west & north. And as an extra, there also exists dragons: white, red and green. And to make it more interesting there is four examples of each tile. In a full set of tiles in Japanese mahjong, there therefore is (9*3+4+3)*4 = 34*4 = 136 tiles.

The goal is the make a complete mahjong hand, which consists of 14 tiles.
Of the 14 tiles, you must make 4 sets (a set is three-of-a-kind or a straight that is 3 tiles long) and one pair. And to make it more difficult, you have to fit your combination of sets so that it matches a pattern with which you are allowed to win. A pattern that allows you to win is called a yaku. In the beginning, there is usually quite difficult to know if you have a yaku or not, but it gets easier after a couple of matches!

You can find the complete rules here.