THE FORK
In this position, the Black Pawn on f7 is attacked twice: once by the White Bishop on c4, and again by the Knight on g5. The Pawn is only protected once, by the Black King. So, White can freely capture this Pawn, but with which piece should he or she capture the Pawn? With the Bishop, or with the Knight?
Looking closely, we notice that the Black Queen and Rook are a Knight's move away from each other, and the focal point is f7! So, White captures the Pawn on f7 with the Knight...
Forking the Queen and the Rook.
A fork occurs when two or more pieces are attacked by a single piece. This usually results in the loss of one of the forked pieces. In this example, for instance, Black will move the Queen to safety at e7, and then the White Knight will capture the Rook.
Every type of piece is capable of delivering a fork.
Here we see the White Bishop on e7 forking Black's Queen and Knight.
Black will be forced to exchange his Queen for White's Bishop and Knight.
Here the White Bishop is forking the Black Queen and Rook. Once again, the Queen will be rescued (by moving it, or by blocking the diagonal by moving the Black Bishop to e7). Either way, the Black Rook will be captured on White's next move.
Here the lowly Pawn takes on a Bishop and Knight, winning one of them.
A Rook can do the same job if placed between two pieces.
When it comes to forking, the Queen can do some amazing things. Here, White appears to be outnumbered materially. But in one move, the lone Queen manages to fork every one of White's pieces...
1. Qd5
The Queen moves to d5, and thanks to her ability to move as a Bishop and a Rook, forks everything! The King must be gotten out of check (by moving it, or by interposing the Knight on f7), which allows the Queen to capture the undefended Rook on White's next move.
THE PIN
White has just moved the Bishop to b5, "pinning" the Knight at c6. A pinned piece is a piece which blocks the attack of a more powerful piece. Here, the Knight is pinned to the King and cannot legally move (as that would involve moving into check).
Since the Black Knight on c6 is the only defender of the Black Pawn on e5, White will be able to capture the Pawn with the Knight on f3 without fear of the Knight being recaptured. That is, unless Black alters the situation...
Black alters the situation by moving the Bishop to g4, pinning White's Knight on f3 to White's Queen. Notice that this pin is slightly different from the first pin we saw. The first pin is an "absolute pin" because Black cannot legally move the pinned Knight. But this pin on the White Knight on f3 is a "relative pin"; there is no rule against White moving the Knight, it's just that the Queen will be captured if White chooses to move the Knight.
Here, the Rook is pinning the Queen to the King. Black cannot move the Queen out of the way, as that would be moving into check. Black will lose the Queen on White's next move.
THE SKEWER
A "skewer", also known as an "x-ray attack", is similar to a pin, except that the more valuable piece is in the front line, attacked directly, and when it moves to safety, the less valuable piece behind it is captured. Here, the Black Queen will move to safety, and the White Bishop will capture the Rook.
DOUBLE ATTACK & DISCOVERED ATTACK
A "Double attack" occurs whenever two pieces attack one piece.
At first glance at this position you might think that the Knight at e6 is under double attack: once from the Black Queen on b6, and again by the Black Pawn at f7.
But, in fact, the f7 Pawn is currently pinned to the King by the White Bishop on h5. Since the Pawn cannot move, the White Knight is really only attacked once: by the Queen. Since the Rook on e1 protects it, the Knight is okay.
But White has a move which will result in a discovered attack, a double-attack, and checkmate...
1. Ng7#
White moves the Knight out of the way of the Rook, revealing or "discovering" an attack against the Black King from the Rook. At the same time, the Knight is now also attacking the King, so it is a double-attack and checkmate. There is no way that Black can block both attacks.