While it is uncertain where exactly chess originated from, the rules for the modern form of chess were first formulated in 16th century Italy. Though modified occasionally until the early 1900s, the game has remained largely in the form it is in today since then.

Chess is a two player game played with 16 pieces each on an 8 x 8 divided board. Each player has eight pawns, two knights, rook and bishops and a king and queen. The goal of the game is to take your opponent's king, while protecting your own.
Players begin by choosing their color, white or black, either by agreement, chance or appointment and setting their pieces on the 64 space board. Each piece is called white or black, but the actual colors of the pieces can vary as long as they are different enough to be distinguished from one another. Likewise, the board traditionally has white and black squares, but can be any combination of contrasting colors.
The board is placed so that a light colored square sits at the near right corner. The pieces are laid out on the nearest 16 square with the row closest to the player set up as follows: the two outside pieces are the rooks, or castles. Just inside the rooks are the knights or horses and inside of these are the bishops. The king and queen take up the two remaining spaces in the back row with the queen residing on her own color. Meaning that the light colored queen always starts on the light colored square and vice versa. The 8 pawns take up the second row.
The player controlling the white pieces always goes first. A common phrase to describe this is: "Fire before smoke". After the initial move by white, players alternate moves. Gameplay continues until there is a draw or either player traps their opponent's king, which is called checkmate.
Each piece moves differently. Rooks move horizontally and vertically as many spaces as the player chooses without jumping over a piece. Bishops move diagonally in the same manner. The queen can move either diagonally like the bishop, or horizontally and vertically like the rook. The knight is the only piece that can jump over another piece. The knight moves in an "L" shape, moving two spaces horizontally or vertically and one space horizontally or vertically, or vice versa. The pawn moves one space at a time, always forward, except when capturing another piece. In this instance it can only capture diagonally.
Aside from these standard moves, the king and rook can perform a special move called "castling". If the king and rook have not moved yet, and all of the spaces in between them are vacant, then the king can move two spaces toward the rook, and the rook can move to the space the king just skipped over. The rook basically jumps over the king to rest beside it. Castling cannot be done while the king is threatened.
A pawn that moves all the way across the board to the opponent's back row becomes a queen. This is called "queening" or "promotion". The player could also choose to take a knight or bishop or rook instead. This is called "underpromotion".
In a tournament or other serious play scenario, if a player touches one of his pieces during his turn, he must move it if he can legally make a move with it. If a player touches one of his opponent's pieces then he must capture that piece if it can be captured. If for some reason a player wishes to move his piece within its own square, to center it, for example, he must let his opponent know of his intentions beforehand. Only the player whose turn it is may do so. These rules are usually somewhat more relaxed in casual play.
If a player makes an illegal move, he must retract it and make another move with the same piece, if possible. If the mistake is noticed afterwards, the game should be reset to that position and restarted. If the game is a speed game where the match is times, the illegal move may cost the player the game, even if it was a simple error.
The game is over when one king is checkmated or the players call a draw, meaning that the player whose move it is not in check but has no legal moves or if there is no possibility for either player to call checkmate.
A game may also be called a draw if 50 moves have been played without a piece being captured or a pawn moved. If the same board position has been repeated three times with the same player to move and all the pieces having the same potential moves, then it could be called a draw as well. A draw can be called if the player to move has no legal move to make. This is called a stalemate. The game can also be called if there is no possibility for either player to call checkmate, or if both players agree to a draw.
When a player makes a move that puts his opponent's king in jeopardy, that king is said to be in check. If this is the case, then the player whose king is in check must make a move that removes that threat, either by directly moving his king, by blocking the threat by placing a piece in the line of the attack or by capturing the piece that is threatening the king. A player can never leave his king in check. This can be done either by moving the king or blocking the threatening piece from attacking the king. Checkmate is called when neither of these can be done. If the king has nowhere to go to escape the check, if all escape routes are blocked by other threatening pieces or the check cannot itself be blocked, then the king is in checkmate and the game is over.
Throughout the game there may be many "checks" on the king. In casual play, the player who puts the king in check usually announces so by saying "check". In tournament or formal play this is considered unnecessary and even a distraction. Any serious player would know that his king was in jeopardy.
