Why study simplicial homotopy groups? The standard definition for simplicial homotopy groups only works for Kan complexes (cf. http://ncatlab.org/nlab/show/simplicial+homotopy+group). I learned that the hard way, when I tried to compute a very simple example, i.e. the homotopy group of the boundary of the standard 2-simplex. 
My naive idea to actually compute simplicial homotopy groups for arbitrary simplicial sets was taking the fibrant replacement. But obviously we need a model structure for that. Then again, a weak equivalence in the usual model structure for simplicial sets is precisely a weak equivalence of the geometric realization.(cf. http://ncatlab.org/nlab/show/model+structure+on+simplicial+sets) 
As I understand it so far, the only satisfactory way to talk about simplicial homotopy groups requires the notion of "classical" homotopy groups. Hence my question: why does it still make sense, to actually talk about simplicial homotopy groups in the first place?
 A: Well, both in the book of Peter May and the paper from Curtis it is proved that the homotopy category of the category of Kan simplicial sets is equivalent to the homotopy category of CW complexes. The equivalence is given by the adjoint functors "geometric realisation" and "total singular simplicial set". This is one of many motivations to study simplicial sets as a very suitable "discrete model" for topological spaces. One very nice thing about simplicial sets is how constructive things are (often :) ). As far as I have read, simplicial sets have been first used for constructing classifying spaces for topological groups.
The fact that we can define the homotopy groups within the category of Kan simplicial sets is great, because this gives us the opportunity to study them using some combinatorial tools as well. You can check this paper
http://arxiv.org/abs/1211.3093 
Here the authors develop a polynomial time algorithm, which computes homotopy groups of simply connected simplicial complexes. For example, starting with a finite simplicial complex model of the 2-sphere, we can introduce an ordering of the vertices and generate a simplicial set. This simplicial set will never be Kan, but using Postnikov towers and a suitable model for Eilenberg-MacLane spaces, one can build up Kan simplicial sets, which approximate it homotopically and use the Postnikov stages for the computations. The result is beautiful and very constructive.
I hope this gives some motivation for the study of simplicial homotopy groups.
A: To compute the homotopy groups of a simplicial set $X$, you need to be able to construct a weak equivalence $X \to Y$ where $Y$ is a Kan complex, and then compute the homotopy groups of $Y$ using the definitions you were discussing.
This might seem circular - you need to detect if $X \to Y$ is an equivalence.  However, you can construct $Y$ directly using certain more elementary equivalences.  Specifically, for a map of a horn $\Lambda \to X$ we can form the pushout of the diagram $\Delta \leftarrow \Lambda \rightarrow X$, called $X'$; on geometric realizations this is homotopy equivalent because you can construct an explicit retraction.  The class of maps $X \to Y$ generated by such pushouts is called the family of anodyne extensions of $X$, and you can always construct an anodyne extension which is a Kan complex by the small object argument (you keep gluing in solutions to every possible horn-filling problem).
If you want a more canonical answer, there is also Kan's $\operatorname{Ex}^\infty$ construction.
If you want another reference, there are Kan's older papers, and my recollection is that Joyal and Tierney has quite a number of details as well.
A: If $X$ is a simplicial set which is not Kan, you can compute the homotopy groups
of $X$ by choosing a weak homotopy equivalence $f: X \rightarrow Y$ where $Y$ is Kan and
then applying the construction you are familiar with to $Y$. There are many ways of characterizing the relationship between $X$ and $Y$ without even mentioning topology or model categories (though I'm not sure it is so helpful to avoid these). For example, you can
take any map $f: X \rightarrow Y$ with the following property: for every Kan complex $Z$,
composition with $f$ induces a bijection from $[Y,Z]$ to $[X,Z]$, where $[K,Z]$ denotes the set of maps from $K$ into $Z$ up to (simplicial) homotopy. There are several purely combinatorial constructions of $Y$ from $X$: for example, Kan's $\operatorname{Ex}^{\infty}$ functor.
A: So this entire discussion is in Goerss and Jardine's "Simplicial homotopy theory" and also in May's "Simplicial objects in algebraic topology". Also Curtis' papers and monographs are very nice and classical. One aesthetic reason that one may want simplicial homotopy groups is to show that we can calculate homotopy groups within the category of simplicial sets. Thus one sets up this machinery. I think that Milnor proved the comparison between simplicial and let's say topological (even though this is not quite accurate, I think something like CGHaus homotopy groups) groups.
Here is how you speak about homotopy groups in the context of simplicial sets: First you need the notion of a horn. A horn is the boundary of an n-simplex with a lost face. Now you must remember that simplicial sets have arrows on their edges, so we have a couple of different horns in each dimension. In dimension 2 for example, we have three different horns (and will need all three of these horns for our to define the fundamental group, in addition to a three dimensional horn to give associativity). So we now define a Kan complex as a simplicial set in which each horn may be filled out to any n-simplex that it is contained in (this is written as a lifting property). 
So we will use this to define the fundamental groups, as the higher homotopy groups are analogous. Pick a basepoint, and two loops based at that point (if we do not want to talk about basepoints this discussion works for fundamental groupoids). This can be realized as a map of a two horn into the simplicial set. Now pick a horn filling (it doesn't matter which, they all differ by homotopy equivalence, namely an even higher horn fills this choice of two horns. This is similar to higher category theory.). The group operation of the two loops is the new loop created on the boundary of the two simplex. Considering the other two horn fillings will give left and right inverses in the group, which must be shown to be homotopy equivalent (by more horn fillings) 
All the properties that you would expect of such composite can be shown to be true by more (generalized) horn fillings. These are called anodyne extensions. It turns out if you can fill all of your horns, you can fill all of your anodynes. This will show that composition is independent of choice of representatives.      
