In chess, double check occurs when the king is placed in check by two pieces at the same time. This can only happen if both checks are given in the same move. For this reason, every double check is also a discovered check - one in which the "discovering" piece also gives check itself.
+----+----+----+----+----+----+----+----+
| | | | | r | | k | |
+----+----+----+----+----+----+----+----+
| | | | | | | p | p |
+----+----+----+----+----+----+----+----+
| | | | | | | | |
+----+----+----+----+----+----+----+----+
| | | | | n | | | |
+----+----+----+----+----+----+----+----+
| | | | | | | | |
+----+----+----+----+----+----+----+----+
| | | | | | | | |
+----+----+----+----+----+----+----+----+
| | | | | | P | P | P |
+----+----+----+----+----+----+----+----+
| | | | | K | | | R |
+----+----+----+----+----+----+----+----+
In this position, with Black to move, the knight on e5 can deliver double check by moving to d3 or f3, when both it and the rook on e8 will be giving check to the White king on e1.
Double checks are particularly nasty to deal with because they restrict the available options so much. Usually, when the king is placed in check, the defender has three options:
- Capture the checking piece
- Interpose a friendly piece between the king and the checking piece
- Move the king
However, in the case of a double check, the first two options are not available. You can't capture two checking pieces in one move, and you can't place any of your own pieces in two lines of fire at once, because to do so would involve placing the piece at the intersection point of the two checks, which is (of course!) where your king is sitting. So the only option is to move the king. In the above position this causes few problems, but consider the following, slightly different position:
+----+----+----+----+----+----+----+----+
| | | | | r | | k | |
+----+----+----+----+----+----+----+----+
| | | | | | | p | p |
+----+----+----+----+----+----+----+----+
| | | | N | | | | |
+----+----+----+----+----+----+----+----+
| | | | | n | | | |
+----+----+----+----+----+----+----+----+
| | | | | | | | |
+----+----+----+----+----+----+----+----+
| | | | | | | | |
+----+----+----+----+----+----+----+----+
| | | | | | P | P | P |
+----+----+----+----+----+----+----+----+
| | | | Q | K | B | | R |
+----+----+----+----+----+----+----+----+
All we have done is add an extra queen and bishop for White, on their starting squares, and (for dramatic effect) a White knight on d6, attacking the Black rook. Black has an enormous material disadvantage, but has the amazing move Nf3++, which is double check and checkmate. This is worth its own diagram:
+----+----+----+----+----+----+----+----+
| | | | | r | | k | |
+----+----+----+----+----+----+----+----+
| | | | | | | p | p |
+----+----+----+----+----+----+----+----+
| | | | N | | | | |
+----+----+----+----+----+----+----+----+
| | | | | | | | |
+----+----+----+----+----+----+----+----+
| | | | | | | | |
+----+----+----+----+----+----+----+----+
| | | | | | n | | |
+----+----+----+----+----+----+----+----+
| | | | | | P | P | P |
+----+----+----+----+----+----+----+----+
| | | | Q | K | B | | R |
+----+----+----+----+----+----+----+----+
Both of the checking pieces are attacked, but neither can be taken because the check from the other piece would still exist. Therefore, White has to move his king, but where to? The queen and bishop block the exit squares to the left and right, and the only other two squares (d2 and e2) are covered by the Black knight and rook respectively. It's checkmate. For another pretty example, see smothered mate.
It's worth noting that Triple Check is not possible in chess. If you can immediately see why this is, then you're a better player than me - I had to sit down for a full half an hour with a chessboard when I was eight years old to try and figure it out. | 
