The weak dark square complex is a group of squares that are undefended by nearby pawns on white squares. For examples in this article, Black has pawns on f7, g6, and h7, and the squares f6, g7,and h6 are undefended by Black pawns. They form a weak dark square complex. The weakness of this complex can be reduced by a dark-squared bishop defending the complex (e.g. a fianchettoed bishop). Conversely, the weakness can be accentuated by trading off Black's dark-squared bishop, especially if White gets to keep his.
In the first example, Black's pawn on h7 is attacked and Black makes the ill-advised decision to protect it by playing 13...g6, blocking the diagonal instead of 13...h6, moving it off the diagonal. This minor decision has major consequences as White exploits the weak dark squares around the king.
In the second example, Black plays the opening well, but in the middlegame, he decides to prevent a move by White that would have been mildly annoying by creating a permanent, serious weakness in his own position that White is able to utilize for the rest of the game.
In the third example, from Grandmaster play, White sacrifices a pawn for the sole purpose of weakening the dark squares around Black's king. Black valiantly tries to cover the weakenesses, but White brilliantly finds a way to penetrate Black's defenses.
The lesson of these three examples is clear: do not create a weak square complex in front of your king and then trade away defenders of those weakenesses. Conversely, if your opponent creates a weak square complex in front of his king, notice it, get rid of defenders, and exploit it. Having your queen and a bishop of the proper color will be extremely helpful in taking advantage of this weakness.