The rules really aren't that complicated.
Your goal is simply to have a hand consisting of four sets and a pair of tiles. Sets count as either having triplets of the same tiles, or a set of three tiles numbered in order. For example, you could have a set of two pin, three pin, four pin. Honour tiles do not have numbers and thus cannot be used in sets.
When you're one tile away from having a proper hand, you can declare riichi. After that you have to continually discard all tiles you draw until someone else plays the tile you need or you draw it yourself.
There's also some more complicated rules involving discards. If someone discards a tile that you would complete a set, you can claim it. and complete your set with it, throwing your tiles in the new set to the side and combining them with the discard. However, in order to win with an open hand like this you must ensure your hand has at least one "yaku". A yaku is a special arrangement of tiles in a hand. For example, having a set of dragon tiles counts as a yaku. Yaku also score you higher points.
http://www.japanesemahjong.net/mahjong-rules/yaku/Finally there is the dora tile. The dora tile is simple. It's a tile that's flipped up at the start of the round. If your final hand has a tile in it that is one above the dora tile, then you score extra points. For example if the dora tile is 3 sou, and your hand contains a 4 sou, then you get extra points.
Practice here and it all makes sense
http://www.gamedesign.jp/flash/mahjong/mahjong_e.html
