The rules
Black moves first. Every move must flank: you place a disc so that one or more of your opponent's discs sit in an unbroken line between the new disc and another of yours. Every disc you flank flips, in all eight directions at once. If you have no flanking move available you pass, and your opponent moves again — you have not lost your turn for good. The game ends when neither player can move, which is frequently before the board is full, and the player with more discs wins.
Why more discs is usually bad
The commonest way to lose Reversi is to win the early game. Every disc you flip is a disc your opponent may flip back, and a big lead in the midgame normally means you have fewer safe squares left to play. When you run out of good moves you are forced to play next to a corner, and that hands the corner over. Corners can never be flipped, so they anchor whole edges for the rest of the game.
That is why our engine weights position and mobility rather than counting discs, and why it often looks like it is losing until the last dozen moves. Move generation is verified against the published position counts for the opening — 4, 12, 56, 244, 1396, 8200, 55092 and 390216 paths at depths one through eight — so the rules, including passing, are not a matter of opinion.
Free review
After every game the review shows what the engine would have played on each of your moves and what the choice cost. Unlimited, no account, no daily cap — the thing the big sites charge for.