A yaku「役」 is a certain pattern in a mahjong hand that scores points, and a necessary condition for winning. They are analogous to hands in poker. Unlike poker, multiple yaku can be combined into one hand. Under standard rules, there are 26 different yaku and 11 yakuman (a special class of high-scoring yaku). In addition, one special case may be allowed in nagashi mangan.
In order to win any hand, the hand must have at least one yaku. Even if a hand has dora, it cannot score without a yaku. Declaring a win without yaku can score a chombo penalty. Therefore, when learning the game, it is essential to learn the different types of yaku.
Each yaku has a specified amount of han「飜」, generally based on their rarity. You can also increase the hand's han count with dora, but this does not count as a yaku. A hand's total han value, counted from adding up yaku and dora, determines the points you'll get in the end.
The most important yaku to learn is riichi, since it is very common and very easy to achieve. A new player may find it easiest to never open their hand, always calling riichi, and get a few wins. However, this strategy is far too simple for serious play; a variety of yaku should be employed.
Any closed hand that reaches tenpai may declare "riichi" to score this yaku. Declaring riichi requires you to bet 1000 points, and announces you are in tenpai.
Have sequences of "123", "456", and "789", all of the same suit. Simply having every tile from 1-9 is not enough; the hand must have those exact sequences.
Three closed triplets (and/or quads). Each of the three triplets must be formed without calling any tiles; a triplet completed by ron does not count as closed. The fourth group can be an open triplet or sequence.
The hand is composed of nothing but all terminals and honors. † This hand may be considered as 4 han, because it is impossible to score this hand without either chiitoitsu or toitoi.
The hand contains two triplets (and/or quads) of dragons, and a pair of the third dragon. † This hand may be considered as 4 han, because it is impossible to score this hand without two sets of yakuhai, from the two dragon triplets.
This hand has one of each of the 13 different terminal and honor tiles, plus one extra terminal or honor tile. Some rules may allow double yakuman for a 13-wait set.
A non-dealer has a winning hand by their first draw. In addition, no player can make tile calls before then.
Special case
This yaku does not specifically deal with a player's hand, but rather the player's discards. In addition, the hand may or may not be at tenpai upon scoring. The result of nagashi mangan modifies the case of ryuukyoku, where point exchanges equate to mangan.
The above list is generally considered as the "standard yaku", including the yakuman. However, the game offers an even larger array of tile patterns. Various house rules may opt to use some of these yaku on their own.
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Agari: 
Agari: 