Trap your opponent's discs between yours to flip them, own the most pieces when the board fills.
Place a disc so that it brackets one or more of your opponent's discs in a straight line between the new disc and another of yours. Every bracketed disc flips to your color. You must flip at least one disc on each move. When neither player can move, whoever has more discs wins.
Reversi has two claimed inventors who could not stand each other. The Englishmen Lewis Waterman and John W. Mollett each said they had devised the game in 1883, and each publicly denounced the other as a fraud; the dispute was never resolved.
The independent record begins shortly after. Its first reliable mention is in The Saturday Review of 21 August 1886, and by 1893 the German publisher Ravensburger was producing it as one of its earliest titles. A New York Times piece in 1895 described it, helpfully, as "something like Go Bang, played with 64 pieces".
The rule that makes it live is the flip. Pieces are never removed and never move; they change allegiance when bracketed, so a position can invert almost completely in a single move, and a board that looks decisively won often is not.
Reversi is the classic two-player disc-flipping strategy game, free to play in your browser with a strong AI plus local and online matches. Built in-house at PlayPendium.