A crossed module is a pair of groups $C$ and $G$, an action of $G$ on $C$, and a homomorphism $\partial: C \to G$ that satisfy 


* $\partial(g\cdot c)=g(\partial c)g^{-1}$, and
* $cc'c^{-1}=(\partial c)\cdot c'$


Let $(X,A)$ be a pointed pair of spaces. Whitehead proved that, in the homotopy long exact sequence of the pair, 
$$
\pi\_2(X,A) \stackrel{\partial}{\rightarrow} \pi\_1(A)
$$
is a crossed module.
Simply put, my question is: what does this give us, other than an extra bit of structure? Does knowing that this is true aid on calculation? Does it aid in distinguishing spaces? Does it give us something really cool that I haven't thought of? (Probably the answer to this one is "yes".)



Crossed module structure on homotopy groups
There are numerous calculations that are easily done with crossed module techniques that are much more difficult to obtain using `traditional' homotopy theory. Some of these use the next stage up, that is crossed squares, and the resulting non-Abelian tensor product. A neat sample calculation is of the homotopy type of the suspension of a K(G,1), if I remember rightly. This is given as the kernel of the commutator map from $G\otimes G$ to $G$. 


You ask is it good at distinguishing spaces. The answer is most decidedly yes. (But I would say that wouldn't I.) MacLane and Whitehead proved that the crossed module models the homotopy 2-type, extending the classification of homotopy 1-types by groups. (Ok there is a price to pay. The correspondence gives 2-types correspond to equivalence classes of crossed modules but the equivalence relation is algebraic not topological in nature so that is reasonable.) Loday proved that homotopy n-types had algebraic models which were crossed n-cubes, n-fold generalisations of crossed modules. 


The cool thing is that conceptually they are one of a linked set of models for low dimensional homotopy information that have geometric significance, and yet are relatively easy to manipulate. I like to say that a crossed module is a normal subgroup that is not a subgroup. Crossed modules 'are' also 2-groups, cat$^1$-groups and various other equivalent formulations.


For some higher dimensional vKT applications, look at higher Hopf formulae in work by Brown and Ellis. 


For applications of crossed modules in non-Abelian cohomology etc. look at Larry Breen's work, or for a gentle introduction, my Menagerie notes which you can find on the n-Lab. 
I could go on listing things but will stop here. If you (or anyone else needs more detail) ask me or ask here.


