Motion of a rigid body: Why is motion only a compostition of two forms: rotation and Translation? If you are in a 2-D space the motion of any rigrid body is composed of 2 different things, namely rotation and translation.
If you are in 3-D space the motion of any rigrid body is also composed of 2 different things, rotation and translation.
If treating the world relativistically you get a 4-D space whre you can also treat the motion of a rigrid body by 2 different things namely rotation and translation.
To me it seems as if the number of "things", which motion is composed of, does not change. 
But why is it 2 and not any other number? Why is there only translation and rotation? Why isn't there a third thing for example "transformation" or something else which also has something that is conserved like momentum and angular momentum? 
Is there some explanation for this?
 A: There is no simple intuitive answer to your question - it follows from various group theoretical theorems: the Euclidean group and Poincaré group decompose to so called semi-direct products of (1) a group of translations and (2) the rest of the isometric transformations. In the Euclidean group, the "rest" turn out to be rotations, and in the Poincaré group the rest are rotations combined with boosts. The crucial decomposition follows from (and is logically equivalent to) the following: if $\tau$ is a translation and $\gamma$ any general isometric transformation, then $\gamma\,\tau\,\gamma^{-1}$ is always also a pure translation. 
You may find, however, some flavour, though, of what you're after in the highly elegant and remarkably simple proof of Euler's rotation theorem (see Wikipedia page of this name). 
Some more mathematical details: your question boils down to two essentially mathematical questions and can be stated thus: 


*

*How do we describe the identity connected component of the Euclidean Group $E^+(N)\cong SE(N)$ of all transformations that preserve the length between any pair of points in $N$-dimensional Euclidean space, i.e. how do we describe the group of isometries of $N$-dimensional Euclidean space which preserve orientation? and

*How do we describe the identity connected component of the Poincaré Group $P^+(N)$ of all transformations that preserve the proper time between any pair of points in $N+1$-dimensional Minkowski spacetime, i.e. how do we describe the group of isometries of $N$-dimensional Minkoswki spacetime which preserve orientation?
and, from the property of length / proper-time preservation alone, it is found that one can wholly define the two Lie groups in question. The fact that we restrict ourselves to the identity connected component of the groups means that we include only transformations that preserve orientation.
In the first case, it can be proven that $SE(N) = T(N) \rtimes SO(N)$, that is, our Euclidean group is the semidirect product of the $N$-dimensional translations with the group of orientation preserving orthogonal transformations, or rotations. This means that every orientation preserving isometry can always be written as a translation followed by a rotation. But note that $N$ dimensional rotations for $N>3$ are quite complicated things: they are essentially matrices of the form $\exp(H)$, where $H=-H^T$ is real and skew-symmetric, so they require $(N-1) N/2$ real numbers to specify them. In 3 dimensions, they need three numbers for specification, so we can think of a rotation about an invariant axis, but in dimensions greater than 3, the dimension of the vector subspace of $N$-dimensional Euclidean space left invariant by the rotation is generally greater than 1, so the axis concept no longer works.
In Minkowski spacetime, you not only have rotations and translations as your basic isometries, but you must also add boosts to your list. Two boosts compose in general to a boost and a rotation, this is the reason for Thomas Precession. 
Again, it is a theorem that our group decomposes to a semidirect product, i.e. $P^+(N) = T(1,\,N) \rtimes SO(1,\,N)$ and we can represent any isometry of Minkowski spacetime as an $N$-dimensional translation followed by an $N$-dimensional Lorentz transformation. The latter further decomposes into a boost followed by a rotation (or rotation followed by boost).
For an idea of what is involved in deriving the description of these groups from the assumption of isometry alone, you can see:


*

*Example 1.3, "Three Dimensional Rotation Group $SO(3)$"

*Example 1.10, "General Classical Groups and The Proper Lorentz Group $SO^+(1,\,3)$"


on the page "Some Examples of Connected Lie Groups" on my website.
A: I maybe seriously underestimating the OP's question (and not quite sure if I understand it) but I believe he/she is simply asking why we only need two quantities (rotation and translation) to describe 3D space.
I'll start with 2 dimensions:
Imagine you're playing the game Asteroids and you want to move your spaceship a little to the right to get out of the way of an incoming asteroid. Now for simplicity's sake, lets do away with momentum, etc. and just pretend that the game is played in steps and you can only move one unit length per turn. (It doesn't matter what the unit length is, it has no bearing on the demonstration). 
For your first move you decided to make your spaceship go one unit to the right. Now if we were plotting the spaceship on a graph a geometrist would say we translated (or slid) our ship along the x-axis one unit. We can slide the ship in any direction (or for the geometrist, along any axis) we want: left (-x), right(x), up(y), or down(-y), but in this case we decided to just go to the right.
Now let's pretend an asteroid is coming toward us from directly above the ship, and we want to steer the spaceship so it can be in a position to fire before we're destroyed! We would then rotate the ship (which costs one move) 90 degrees toward the asteroid, and since we only get one unit per turn, we can't slide anywhere.
You can see that by just using translation/rotation you can play a simple 2D game like Asteroids.

But what happens in 3 dimensions? Can we only use rotation and translation to navigate around?
Let's go back to the spaceship, but this time we'll pretend we're actually inside the ship so we can see if rotation/translation will work to move around.
For the first move we decided to go to the right (now it's forward) one unit, and then for our second we needed to turn the ship (rotate upward) and shoot the asteroid. Again we're playing in steps, do don't worry about your momentum being conserved or anything like that.
Now that we destroyed the asteroid our commander decides to call us back to our home planet where we can refuel and rearm our weapons. But when we look out the window we notice the planet is not only in front (x-axis) and above us (y-axis) , but it is also to our left!
But how are we going to get there? Well, to do that we're going to have to use another axis, the z-axis. The z-axis will let us have two additional directions, left and right. So now not only can we go forward/backward (x-axis) and up/down (y-axis), we can also go left and right (z-axis)!
So do we move along the z-axis? Can we slide? Yes, but even better, we can also rotate! So for our turn, we'll rotate a little to the left (z-axis), and a little upward (y axis) so we can fly directly to our home planet. 

If you do some thinking, you'll realize any point in 3D space can be moved to using just rotation and translation. I highly suggest you read the Euler rotational theorem Rod Vance posted above, it's got some great visuals to help explain his thinking.
Also, in the question you mentioned 4D space, but most often when people are talking about that extra dimension they mean time, and it completes the Minkowski space-time manifold of general relativity.
Sometimes people also reference more spacial dimensions (particularly in linear algebra), but that's quite a bit over my head.
Pictures are from Asteroids (Atari, 1979) and Elite: Dangerous (Frontier Developments, 2014)
A: No matter how many spacial dimensions, there is only one kind of line.  Two points describe a line and between them is only one dimension. 
For rotation, again it doesn't matter which way it's rotating, but the number od possibilities increases with dimensions.  The question is whether any combination of rotating (instantaneous) can be combined to a single result.  
So you still have 


*

*translation. Two points to indicate direction and magnitude. 

*rotation. Two points to indicate axis and magnitude. 


You need more scalar values to indicate the points, but still only have 4 points.
Now it gets funny... for 2D the only axis is not in the space, so two points can't represent a direction.  You need (implied: only axis) and scalar magnitude. 
In 4D and other even numbers, you need a different way to specify what the axis is. But there is only one at an instant, and the magnitude is a single number.
In spacetime, you need to specify the translation (relative to known things) and then can describe the axis of rotation and its rate by assuming the world line of that object.

The number of things is two  because you specified 2 things in advance, and the interesting thing is that you can only have one of each.
Are there more things that might vary by dimension? You need a thing that uses more than 1 dimension (picked from those available), so it could not exist in lower dimensions. 
If you limit yourself to instantaneous attributes,  I don't think there is anything.  If you allow higher derivatives then you can combine multiple motions and rates. For example podholing and precession are interesting things.
