The Alekhine Defense has always been the defense that has puzzled me the most. I thought that the goals of the opening were to occupy and control the center while developing my pieces as quickly as possible. Against the Alekhine Defense I could do all of this while chasing black's knight around the board. How can it be beneficial for Black? This is a fine example of the complexity of the game of chess.
Alekhine's Defense 1.e4-Nf6, is as you have no doubt guessed, named for Alexander Alekhine and he used it quite effectively. It is a hyper-modern defense played against the most common e4 opening. Black looks to allow white to chase his knight around the board while gaining pawn moves that will control the center of the board. In exchange for center control, black will look to undermine the overextended pawn moves by white. However, black cannot play passively and needs to attack the center that white has built up or he will be crushed by the pressure white has built up. Take a look at this game:
 
 

 
Now this is not a typical Alekhine's defense game but it does demonstrate its potential against those not familiar with it.
 
Alekhine's Defense 1. e4-Nf6 and after the most commonly played second move 2.e5-Nd5, the two main lines are either the Exchange Variation or the Four Pawns Attack. The Exchange Variation continues 3.d4-d6 4.c4-Nb6 5.exd6. White has some space advantage but black can work on the half-open center with g6, Bg7 with Bg4 eventually being played. The Four Pawns Attack continues 3.d4-d6 4.c4-Nb6 5.f4. White has larger space advantage but the center is not full established and black has a number of options. Black can play Qd7 with 0-0-0 and f6 putting pressure on white's d pawn. Black can play Nb4 with c5 hoping to exchange the d pawn. Black can play Be7 with 0-0 and f6 attacking the center. Since I cannot post multiple created games in one post here is the algebraic notation.
 
Four Pawns Attack.
1.e4-Nf6
2.e5-Nd5
3.d4-d6
4.c4-Nb6
5.f4
 
Exchange Variation
1.e4-Nf6
2.e5-Nd5
3.d4-d6
4.c4-Nb6
5.exd6