How to win: preventing the opponent's king from moving while attacking it. This is a checkmate.
That's the goal of the game. The rest of the rules just relate to how each piece moves.
King: Can move one square in any direction.
Queen: Can move any number of squares in any direction, but can't jump over pieces in the way.
Bishop: Can move any number of squares along any diagonal it's on, but can't jump over pieces in the way.
Rook: Can move any number of squares horizontally or vertically, but can't jump over pieces in the way.
Knight: Can move in an L-shape. 2 squares horizontally and one square vertically, or 2 squares vertically and one square horizontally. It can jump over any pieces in the way.
Pawn: Can move one square forward, and has the additional option of moving 2 squares forward if it's on its starting square. However, it captures enemy pieces by moving one square diagonally in either direction. The pawn is unique in that it captures and moves differently (unlike all other pieces). In addition, if the pawn reaches the end of the board it turns into a queen, rook, bishop, or knight (your choice).
Also, the game could end where no one wins, a tie or draw. This can happen in a variety of ways:
- Both sides only have a king left.
- Neither side has any material left that's sufficient to possibly make a checkmate. Any example is one side having a king and knight vs a lone enemy king.
- A position has been repeated three times. This is called a three-fold repetition.
- One player offers a draw and the opponent accepts.
- Stalemate: one side has no moves left because his king is trapped and the rest of his pieces have either been captured off the board or cannot move. This is very similar to a checkmate, except that the trapped king is not being attacked.
I've left out some rules, but this is the meat of it.