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There are 4 of each tile in a mahjong set.
There are 3 suits with 9 tiles each -- dots, bamboos, and characters making up the majority of the set (3 * 9 * 4 = 108 tiles).
There are also 7 kinds of honor tile: 4 winds and 3 dragons. (7 * 4 = 28)
Thus, a complete set of Mahjong tiles has 136 tiles: 108 numbered and suited, 28 honor tiles.
A complete mahjong hand consists of 14 tiles. To finish a round, a player needs 4 sets of 3 tiles each along with a pair. A set can take two forms: it can be 3 identical tiles, called pong (example: 3 bamboo, 3 bamboo, 3 bamboo, or red dragon, red dragon, red dragon) or it can be 3 same-suited tiles in a straight, called chii (example: 3 bamboo, 4 bamboo, 5 bamboo; 7 dot, 8 dot, 9 dot).
A special kind of set is when a player has 4 of a tile. called kong. One or more kongs generally yield bonus points. The 4th tile acts as a kind of an extra tile in a hand; after declaring a kong, the player draws an extra tile, and then discards; for the sake of hand composition (four sets of 3, and a pair), a kong is treated as a set of 3 and the extra tile is ignored.
It is not enough to only have a hand of 14 tiles to score points in mahjong: a player must also form patterns, which will be discussed later.
Play goes in a counter-clockwise order. Starting with the dealer, and unlike with a real compass, the seats are labelled as East, South, West, and North. The dealer begins with 14 tiles and all other players have 13 tiles. The first turn of the round ends with the dealer discarding a tile.
On every turn until the end of the round, players hands are kept at 13 tiles. After a player discards a tile, several possibilities emerge:
Otherwise, the discarded tile is now dead, off limits for the rest of the game, and the next player draws a tile.
Whether a player starts their turn by chii, pong, kong, or drawing a tile, they end their turn by discarding a tile, at which point this cycle repeats.
This is the biggest different between various kinds of mahjong.
One common trend is for a hand fully consisting of 1 suit being more valuable than a hand of mixed suits (a flush), as it's more difficult to compose; another is for sets containing honor tiles being more valuable than sets of numbered tiles, as honor tiles can only be matched in identical sets ("pong") and cannot make up straights ("chii").
A hand consisting entirely of pongs/kongs along with the pair (3 or 4 of a kind) is usually worth a fair amount; a hand consisting entirely of chiis is worth a small amount, while a full straight flush (123, 456, 789 in a single suit) is generally worth more. Repeating chiis (example: 2-3-4 dot, 2-3-4 dot) is also usually worth more points.
Why is a "pong" valued more than a "chii"? Consider two cases: a player has a 4 of bamboo and 5 of bamboo and wants to make a set. There are 8 possible tiles to form the set: the 3 of bamboos (four copies) or the 6 of bamboos (four copies) can be used, giving 8 potential tiles to make the set. Now consider someone with 2 of the red dragon: it can only possibly be matched with another red dragon, of which only 2 exist. This makes a "chii" roughly 4 times easier to make than a "pong" !
If a player can make a successful hand without taking others' discards (calling chii / pong / kong), they have what is referred to as a "closed hand", which is generally more valuable than an "open hand" (a hand including tiles taken from others' discards). The inversion of this case is if another player discards the final tile to complete someone's hand: this is sometimes worth more points, and punishes the discarder. In a self-drawn win, the other 3 players pay the winner equally.
This may be combined somewhat freely. Read this article for more details

Fewer patterns than in Zung Jung but reading beyond the scope of this guide is necessary to fully understand the intricacies of this strange Chinese domino puzzle.
