Now I'm curious. Where would one go to learn how to play Mahjong?
TV Tropes has an
article on it that gives a decent basic introduction.
If the dealer wins a hand (or, as I understand it, wins at least 1500 on Tenpai), you go into a "bonus" hand, and the deal doesn't advance. After 5 bonus hands, (can be on different players' deals) the requirement to declare a win increases to 2 han.
The standard rule on renchan ("bonus hands" where the dealer doesn't rotate) is that it occurs if the hand is drawn or the dealer wins the hand. Japanese variants usually have the exception that the dealer still rotates (but the honba, or bonus hand number, still increments) if the dealer is no-ten in the event of a draw by running out of tiles. House rules vary for abortive draws, although kyuushuu kyuuhai is virtually always a renchan since it's optional to declare.
I've noticed Touhou Unreal Mahjong doesn't stick to the standard rules, though.
Since Rinshankaihou was mentioned with no explanation to what it actually is, I'll elaborate on it here.
Rinshankaihou is a single yaku that is attained when you tsumo on a dead wall draw from your self-declared or called kan. It is NOT a chankan, where you win off a tile that is used for someone else's kan. Since it's a tsumo, it automatically follows that you'll get yaku for Menzentsumo. In popular media, Saki from the anime of the same name is known for using this technique to get her big hands.
Actually, rinshankaihou does NOT necessarily imply menzen tsumo. Saki just gets both simultaneously a lot.
Menzen tsumo means you drew ALL tiles yourself, including the winning tile. (Menzen = no stolen discards, , tsumo = draw winning tile yourself.)
The only way to get both is to have a tenpai hand with no stolen discards and at least one an-kotsu (i.e. concealed equivalent of a pon), draw the 4th tile to declare a concealed kan on your an-kotsu, and get rinshankaihou from that.
If you stole a discard earlier, then kan and draw the winning tile from the dead wall to win by tsumo, you only get rinshankaihou, not menzen tsumo. Same goes in any scenario where the kan is open (i.e. if you convert a pon to a kan, or if you have three of the same tile and call kan when someone else discards the fourth).