Looking for a translator and an editor to work with me, because I'm bad at this translating thing. PM me on IRC or something.
Roughly half of the translation was copied/edited from Underwater-Mahjong's subs for the anime series. Subsequent releases may be a few days after airing so I can make the encode look less bad.
Most terminology is left untranslated because I'm a complete weeb^W^W^W^W these are the most widely understood terms that don't involve unholy combinations of Japanese, Cantonese and English words. Abbreviations like メンホン are half-translated ("closed hon'itsu") since I don't really expect you to be familiar with the Japanese abbreviations, but I do expect you to know all the yaku names and their individual abbreviations ("hon'itsu" and the full "hon'iisou"). Even if you aren't familiar with that, here's a quick summary. Note that these are simplified descriptions of the rules and exclude many details.
A mahjong game can be played in multiple formats, typically East-only (tonpuusen) or East+South (hanchan). A tonpuusen nominally consists of the four hands East 1 through East 4, while a hanchan includes South 1 through South 4 as well. Play proceeds counterclockwise, with the dealer rotating counterclockwise after each hand. If the dealer for the current hand wins, the hand "repeats" and is considered a bonus round.
A player's hand consists of 13 tiles, and the goal is to be able to arrange the tiles into four groups and a pair by either drawing a tile (self-draw, or "tsumo") or winning off another player ("ron"), where a group may be a triplet, a quadruplet, or a sequence. (Note that a quadruplet is "effectively" only three tiles in the hand. We'll get back to this.) Points are awarded based on how many predefined patterns ("yaku") the hand satisfies, how many bonus tiles are in the hand ("dora") as well as the structure/shape of the hand itself. Yaku and dora contribute to the "han" portion of the score calculation, while the shape contributes to the "fu" part. (tl;dr: more han = more score. and also more fu = more score.)
Most hands score either 20 or 30 fu, which is all you need to know if you're playing Japanese mahjong as an amateur, but this series really digs into the less-common parts of the scoring system. Tiles can be divided into four categories: the "simples" (numbered tiles from 2 to 8), the "terminals" (numbered tiles which are 1 or 9), the "winds" (east "ton", south "nan", west "shaa", and north "pei"), and the "elements" (white "haku", green "hatsu", and red "chun"; very often mistranslated as "dragons"). The wind and element tiles are also collectively referred to as "honours".
A sequence never adds any fu, a triplet adds some fu, and a quadruplet adds even more fu. A triplet/quad adds more fu when concealed than when open, and the fu amount also increases if it is of a terminal or honour rather than a simple. In addition, the pair contributes to the fu if (i) it is of the player's seat wind or the prevailing wind or (ii) it is an element. (tl;dr: quadruplet of terminal/honour = lots of fu. lots of quadruplets/triplets = even more fu.)
Apart from the shape, how the hand is won can also contribute to the fu calculation. Winning on a two-sided wait to complete a sequence gives 0 fu; winning on a two-sided wait to extend a pair into a triplet also gives 0 fu; any other kind of wait grants 2 fu.
Different yaku confer different amounts of han. The simplest and most common yaku are menzen tsumo (concealed self-draw), pinfu (four sequences and no fu points), tan'yao (all simples) and riichi, which are all 1 han. More expensive ones include sanshoku doujun (three sequences of the same numbers in three different suits), which is worth 2 han when the hand is closed but only 1 han when the hand is open, and chin'iisou (all tiles of the same suit), which is worth 6 han when closed and 5 when open. Each dora tile also contributes 1 han, which stacks if a tile can be considered a dora in multiple ways. (tl;dr: more dora good, more yaku also good.)
Among the yaku, there are a few that are situational and don't depend on the patterns within one's hand. Some of these are menzen tsumo ("tsumo" for short), riichi, ippatsu, rinshan kaihou ("rinshan"), chankan, haitei raoyue ("haitei") and houtei raoyui ("houtei"). A menzen tsumo requires the hand to be fully concealed (and as the name implies, must be won with a tsumo, not a ron); a riichi requires placing a 1000-point bet on the table and locks the hand; an ippatsu is scored by winning on the first set of turns after declaring riichi; a rinshan is scored by winning with draw from the dead wall (if the hand is closed, this stacks with menzen tsumo); a chankan is a ron against another player upgrading an open triplet to a quadruplet ("kakan"/"shouminkan"; see next paragraph); a haitei is a tsumo with the last draw; a houtei is a ron on the last discard. These are worth 1 han.
It's worth elaborating on the rinshan kaihou because that's Saki's shtick, but to do that, we first have to talk about parallel universes^W^W how quadruplets work in mahjong. There are three ways to declare a quad: closed ("ankan"), added ("kakan", "shouminkan"), and open ("daiminkan"). Essentially, you set aside the four tiles forming the quad, draw a replacement tile from the dead wall, then either declare a tsumo or make a discard. If you declare tsumo, that's a rinshan kaihou. The dead wall is a section of 14 tiles set apart from the main wall at the very beginning of the hand when players draw their starting tiles, and the only way to draw from the dead wall (and hence win with rinshan) is to call kan. (tl;dr rinshan kaihou is rare and super cool.)
Sometimes a 13-tile hand can be completed (as in, become four groups and a pair) with two or more different tiles, and sometimes the different winning options lead to different hand shapes and thus different scores. A cheaper win is called "yasume" and a more expensive win is called "takame"; a player may choose to ignore a yasume win if they'd rather get more points with the takame. The most common situations where a yasume/takame distinction can happen are with sanshoku, ittsuu, yakuhai, and dora.
This in turn is related to the furiten rule, which is kind of complicated and makes the distinction between having a complete hand shape (four groups and a pair) and going out (four groups and a pair, and any yaku). If a tile gets discarded that completes your hand shape within the current go-around and you don't call ron (either you don't want to or you can't), you can't ron another player within the same set of turns. If you pass on completing your hand with a tile you draw, you can't ron other players for the rest of the round (until you change the shape of your hand). This state of not being able to win off other players is called "furiten".
For hands up to 4 han, the value increases exponentially with the han value and linearly with the fu value, but for 5 han and above, the scoring depends solely on the han value. 5 han hands ("mangan") have a base score of 2000, 6-7 han hands ("haneman") have a base score of 3000, 8-10 han hands ("baiman") have a base score of 4000, 11-12 han hands ("sanbaiman") have a base score of 6000, and 13+ han hands ("yakuman") have a base score of 8000. For a dealer tsumo, the other three players pay twice the base score each; for a nondealer tsumo, the dealer pays twice the base score and the other players pay the base score. For a dealer ron, the victim pays six times the base score; for a nondealer ron, the victim pays four times the base score. (tl;dr: dealer pays more if they lose, but also wins more if they win.)
The amount to pay is declared after saying ron or tsumo. In the case of a ron, this is a single number. Easy. In the case of a dealer tsumo, the three other players pay the same amount, so this is announced as, for example, 4000 all. In the case of a non-dealer tsumo, two numbers are announced, as in 2000 / 4000. These are rendered with a slash between the numbers in the subs, and usually the smaller number is announced first. Sometimes they don't announce the numeric value at all for mangan and above, just "mangan", "haneman", etc., and even for the numeric values, they're quite often abbreviated (e.g. 3900 as "zan-ku", literally three-nine). Abbreviated numeric values will be expanded into the full value in the subs, but "mangan" and so on stay.
Players start with 25000 points, and at the end of the game, a winner is chosen among the four players as the one with the highest score. For the other three players, subtract 30000 points (the "break-even" amount), divide by 1000, and then round to the nearest integer to get a "plus-minus" score. The winner's plus-minus score is the negative of the sum of the other three players' plus-minus score, so that the sum of all four players' plus-minus scores is always zero. In effect, the winner gets a +15 bonus while the other three players get a −5 penalty, which also translates to a 20000-point relative bonus for the winner. Note that many online mahjong games (gamedesign, Tenhou) use different scoring systems. A final score that is neither positive nor negative is simply called "plus-minus zero", or rather, "pura-mai zero" for maximum weeb effect.
tl;dr: Mahjong is hella complicated.