From The Othello Wiki Book Project

Standard othello is a two player game, played on an 8x8 board. Disks are one colour on each side, usually black and white.
The board is set up as shown:

Each player is assigned a colour, and they alternate turns placing one disk (their colour up) on the board at a time.

The aim of the game is to have the most pieces of your colour by the end of the game.

Legal Moves

Moves can only be made to empty squares, and must flip at least one disk.
Disks can be flipped vertically, horizontally, or diagonally.

Any unbroken lines of the opponent's disks between the disk you make your move with and your existing disks on the board are flipped to your colour.
An example:
  
Placing a black disk in the blue square will flip disks in two directions.

Passing

If you have no legal moves you must pass, and it is the opponent's turn again. You must give them a disk if they require one to make their move.

Ending the game

The game ends when neither player has legal moves. This is usually when all squares are occupied, but can also happen when neither player has access to the remaining empty square(s).
Whichever player has more disks of their colour at this point wins the game.
Both players having an equal number of disks results in a draw.