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Reduction Methods:

The basic premise of reduction methods is that it is much faster to solve a 4x4x4 cube (or really any big cube) once it has been reduced to a 3x3x3 cube. What does that mean? A cube has been reduced to a 3x3x3 when the centers have been paired together, the edges have been paired together, and you don't have parity. Basically, a big cube has been reduced to a 3x3x3 when you can solve it with no knowledge of any methods beyond 3x3x3 methods, and you can solve it like a 3x3x3.

There are many strategies for doing this.

The simplest method is just standard reduction (often referred to as redux). First, you pair the centers (make sure the 4 center pieces on each face are the same color). The real difficulty with this on the 4x4x4, or really any even numbered big cube is that the centers are not fixed relative to each other, so you need to make sure that you're solving the centers correctly, or you will not be able to solve the cube. After solving the centers, you pair up the wings to make edges. Fix any parity issues you have, then solve the cube like a 3x3x3.

Redux can be improved in a few ways, but most of them are just different ways of ordering the steps. For example, in Yau, which is the method used to set the current world record, you pair 3 of the cross edges while pairing 2 of the centers, and you pair the remaining 4 afterwards. This leads to a much better set up for a CFOP finish. Z4 is a modified version of reduction in which edges are oriented as they are paired, which leads to a more ZZ like finish.

There's one other way to improve Reduction, but I'll get to it later.

Method List:

Redux

Yau (I use this now!)

Hoya

Z4