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On the variety of 4x4 approaches

Introduction

The regular 3x3 has lots of different ways to be solved. With this in mind, it may sound logical that his big brother has even more ways to be approached! Therefore, I shall only discuss some unique aspects which are typical for a 4x4, compared to a 3x3.

Parity, why does it happen?

If you have solved a 4x4 before with the most commonly used method (the 3x3 reduction), than you've probably already encountered the concept of parity. In mathematics, parity literally means 'to be even or odd'. Applied to cubic puzzles: does it have an even number of layers or an odd number of layers? If you reduce a 4x4 to a 3x3, you're reducing an even-layered puzzle to an odd-layered puzzle. In other words, you're changing its parity. This also explains why one doesn't have any parity issues when a 4x4 is reduced to a 2x2, since you're reducing from even to even.
If you change a puzzle's parity, you're forcing it to change its 'settings'. In other words, you want the puzzle to be something it can't possibly be. In the case of the 4x4 to 3x3 reduction: the puzzle responds by giving situations which are impossible to encounter when solving a 3x3. These are (1) 1 edge oriented wrong and (2) a wrong permutation of 2 edges. The only way to get rid of a particular parity problem, is by doing an algorithm which uses the moves typical for a 4x4 and can't happen on a 3x3 (using the fact that the reduced center pieces and edges are split).

This also applies on other kinds of twisty puzzles. Every time you encounter a parity problem, it means you've reduced the puzzle by pretending it's something else. In this context, parity has a broader meaning, because to be even or odd isn't always clear.
This may seem to be something bad, but it certainly isn't! A 'false reduction' is sometimes just easier (or quicker). Nevertheless, it's always fun to find ways to solve puzzles without changing its parity.

Different methods

3x3 reduction and speedsolving variations

One of the most commonly used methods to solve a 4x4, is reducing it to a 3x3. This means that every set of 4 center pieces and every set of 2 edge pieces are matched up. After all this is done, you practically have a 3x3, which can be solved (by most people who want to solve a 4x4).
The two kinds of parity situations that comes with this approach are explained in the paragraph above, and why it happens will be explained in the next one. (To lift a corner of the veil; it has nothing to do with edge orientation when you reduce the edges.)
This approach is used by the majority of speedcubers. Many tricks and methods have been developed to reduces centers and edges as efficient and as quickly as possible.
The algorithms they use to get rid of parity problems are designed to be fast and simple when it comes to execution. The reason why they 'choose' to have a chance to get a parity problem by using this method, is because it's still a lot quicker than other methods. After all, it's only one (or two) algorithm(s) extra. And despite this, there are even ways to reduce it as such, that you can avoid to have a parity problem!

2x2 (AI) reduction

Reducing a 4x4 to a 2x2 is a parity-problem-free way to solve a 4x4. The reason for this is explained above; you're reducing an even-layered puzzle to an even-layered puzzle, meaning: you're not changing its parity. This approach consists of reducing 8 corner blocks by matching up corresponding pieces. This is a lot harder than the previous method, because the unpredictability and the 'thinking several steps ahead' isn't that easy when you're not used to it. There's more to keep in mind and a true understanding of the ways to shuffle pieces around without affecting others is needed.

This 2x2 (or AI) reduction method shows us a behind-the-scenes feature of the 4x4. A 4x4 is namely a set of 8 'extended' corner blocks, consisting of 7 pieces each. When we treat a 4x4 like this, and thus reducing it to a 2x2, we don't get any parity problems. And this is exactly the reason why you DO get parity problems when you reduce it to a 3x3. Doing this, you match up edges like they belong to each other, while it's in essence the edges who belong to a specific corner!

Layer by layer

Another fun way to solve a 4x4 is by solving it layer by layer. The name of this method speaks for itself: you make the four layers, each at a time, working your way from bottom to top. This is also a fairly easy way to solve it, since it doesn't require that much strategy. Besides, the fact that this is an even-layered puzzle makes the last layer simple to solve, since there aren't any middle edge pieces which decide the kind and orientation of that particular edge.
Solving a puzzle each layer at a time can be applied to a whole range of twisty puzzles, not only standard NxN ones. Getting to know this method using a 4x4 is a good start to dive into this.

This is also a parity-problem-free way to solve the puzzle. The reason for this is you're practically reducing the puzzle as it should look like. In other words: you're not reducing it to another puzzle (with or without another parity), but you use every characteristic of the 4x4 there is and by doing this, you're not changing its settings so you can't have parity issues if you solve it layer by layer.

Combinations

Every method mentioned above (and every method I didn't go through) can be combined to something whole new. For example, you can solve it layer by layer until half the cube is solved and use AI for the remaining 2 layers (4 corner blocks). A variant of this variant is solving the last corner block with a variation of the layer by layer approach (by sliding center pieces into correct position), which makes it easier. Either way, combining two methods with no potential for parity problems results in a parity free solve.
Experiment, see what fits you the most and don't be afraid to challenge yourself and try something new!

4x4 modifications

There are a lot of 4x4 variations and modifications on the Market and every single one of them can be solved using of the methods described above. Of course, this might need some modification. For example, the Master Axis Cube has specificity when it comes to center orientation, meaning that the layer by layer and the AI approach won't go as fast as on a standard 4x4. Nevertheless, it's always fun to approach a puzzle in your collection in a whole new different way. I solved my Master Axis Cube with a 3x3 reduction for a long time, but I did
n't like the parity algorithms which don't mess with center orientation. So I wanted to approach it in another way. Solving it layer by layer or with an AI style was a fun extension to the known capacities of this puzzle.

Every derivative of the 4x4 has these characteristics, unless you're encountering any form of bandaging or jumbling etc. I'll rephrase: every puzzle which acts, turns and can be approached as a 4x4, can be solved in many different ways.
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