American mahjong has a reputation for looking complicated, but the core loop is simple once you have played a few hands. Here is the short version.
The goal
Each of the four players tries to build a winning hand of 14 tiles that matches one of the hand patterns listed on the current year's official card. Hands are built from tile groups like runs, sets of identical tiles, and special combinations named on the card.
The basic turn
Play moves counter-clockwise. On your turn, you draw a tile, then either keep it and discard one from your hand, or pass. Before your own draw, you can also call a tile another player just discarded if it completes an exposed set you are building, then discard in return.
The Charleston
Before the hand begins, players pass three tiles at a time to their neighbors in a set sequence called the Charleston. It is how players trade away tiles they do not need for ones that fit their hand, and it is worth learning early since it shapes the entire hand.
Winning
The first player to complete a valid hand from the card calls 'Mahjong' and lays their tiles down. Scoring is based on which hand pattern was completed and whether it was built concealed or with exposed calls.
The rules above cover the basics — the fastest way to actually learn is to play a few hands with someone who already knows the game.