GambitDeclined.com - The Chess Resource

Winning

To win a game of chess you must trap your opponent's King. When the King is trapped, he is said to be in Checkmate and the game ends.

Checkmate is similar to 'check' however, from checkmate there is no escape. A King is in checkmate if all three of the following conditions are true:

The player cannot capture the piece attacking his King.
The player cannot interpose a man between the attacking piece and King.
The player cannot move his King out of check.

So, c heckmate occurs when the King is in check and he cannot escape.

Checkmate

Checkmate

The split-diagram on the left illustrates two separate examples of checkmate.

The lower example shows that the White King is in check by the Black Rook on b1.

The White King cannot capture the Black Rook as it is protected by the Knight on c3 and capture of the Rook would move the King into another check - moving into check is illegal.

The only squares the King could move to are protected by the two Black pieces - a1 is guarded by the Knight and b1 by the Rook.

With no way out the White King is checkmated and Black wins.

In the upper example, the Black King is under attack from the White Rook at d8.

To get out of check, the only option for Black would be to

get off the eight rank. Unfortunately, the only escape root, the seventh rank, is guarded by White's Rook on e7.

No escape for the Black King means checkmate and White wins.

Other Ways to End

Checkmate is the nicest way to finish a game, it's your goal and it's decisive. However, there are a couple of other ways for a chess game to end.

A player runs out of time - timed chess games give each player a set time in which to win the game. When it's your turn to move, your time counts down. When you have made your move, your opponent's time counts down. When there is no time left on a player's clock, that player loses.
A player resigns - experienced players will foresee a checkmate scenario and preempt it by tipping over his King as a mark of his resignation.

In chess, there are ways to end a game where neither player wins.

A draw by agreement - Either player can offer a draw at any point during the game.
Stalemate - A stalemate exists if although a player is not currently in check, the only legal move open to that player would take the King into check - and moving into check is illegal. In a stalemate situation the outcome is a draw.
Insufficient Mating Material - Self explanatory, neither player has sufficient material to force a mate - for example, when each player possesses only a King each. Such a situation produces a draw.
Triple Occurrence of Position - The game is drawn, upon a claim by a player on the move, when the same position, for at least the third time, (not necessary by repetition of moves) is about to appear.
Fifty-Move Law - A draw is declared if no pawns are moved and no captures are made in fifty consecutive moves.