The discovered attack occurs when a piece is able to attack an enemy piece once a friendly piece moves out of the way. It's not the greatest definition ever (I may have been tired or something when I wrote it), so let's take a look at the principle in the form of a diagram:

Of course, as the Bishop cannot capture the Rook until the next move, Black could easily block the attack with ...c6.

There are several ways in which the discovered attack can be made more powerful; one of these is the double attack, in which the piece moving out of the way also makes a threat. An example is present in the above diagram: should White move his f3-Knight to either e5 or d2, it would threaten the c4-pawn. Now, when Black plays ...c6 to shield the Rook from the Bishop, White can win the c4-pawn with Nxc4.

The discovered check is another type of discovered attack; here, the target piece is the King. A bare-bones example of a discovered check can be seen in the next diagram:

Here, when the e2-Bishop (on the yellow square) moves, the e1-rook will carry out a discovered check on the king on e8. 1.Bg4+ Kd8 2. Bxc8 Kxc8 would win the Black Rook for a Bishop; however, there is something better, as we shall see in the next paragraph.

The double check is the most powerful discovered attack. This is the same as the discovered check, except that the moving piece also gives check. An example from the above diagram would be 1. Bb5++. Now, 1...Kd8 is forced, with mate to follow. 1...Be7 is impossible due to the Bishop check, as is 1...c6 due to the Rook check. This means that when a player is double-checked, he must move his King, as it is the only way in which both checks can be alleviated.

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