There are many rules variations around the world. At Mahjong Night we are presently learning and playing the the Japanese variant known as Riichi Mahjong.
Players begin with a random hand of 13 tiles. They take turns taking and discarding tiles one at a time until one player completes a "winning hand".
A complete hand consists of 14 tiles. This means it includes the most recently drawn tile before a player has discarded. A complete hand consists of four melds and one pair. A complete hand cannot win alone, it also requires a special feature called a yaku. There are many different yaku but some are more common than others and beginners can play effectiely even if they know just the most common yaku.
Some yaku are simply features of the tiles themselves. For example have a meld of the tile called the "red dragon" as one of the four melds in your hand grants the hand yaku.
Other yaku are situational, independent of the contents of the tiles. In other words, sometimes a simple "complete hand" has yaku purely because of the timing when the player achieved it. The most dramatic example would be a yaku called Heaven's Official Blessing. This occurs if the player's initial hand, prior to any discards is a complete hand.
Riichi is a yaku acheived by choosing to gamble on the outcome of upcoming turns. When a player's hand is still concealed, and they are only one tile short of having a complete hand, they may declare riichi by making a 1000 point bet that they will fully complete their hand before anyone else. They place a marker representing 1000 points in front of their discard area. From then on they must discard any new tile they draw unless it completed their hand. If they fail to win this way, the 1000 points will be added to the score of whichever player does win. This feature is distinctive of the Japanese rules of Mahjong and gives it its name: Riichi Mahjong.
Riichi Wiki : Comprehensive English Reference Site about Japanese Mahjong.
Mahjong Guide : Blog with lots of articles. Not sure if it's still active.
Hopeless Girl : Someone's personal Mahjong Blog with some intermediate or advanced strategy tips.
Mahjong Soul : Online multiplayer Mahjong game with anime avatars to spend money on.
Tenhou : Japanese online multiplayer Mahjong.
Majong Soft : Play online against bots. Not a particularly friendly interface but at the end of a hand it provides a really thorough break-down of how the hand was scored.
Riichi Scoring Trainer : Test yourself counting Fu and Han.
Efficiency Trainer : I really like this trainer, helps just practice choosing tiles to discard.