I used to think that the entire theory was intellectual masturbation, but two examples in particular completely changed my mind.
The first is the Pontryagin-Thom construction, which exhibits an isomorphism between the $k$th stable homotopy group $\pi_{n+k}(S^n)$ and the framed cobordism group of smooth $k$-manifolds. This is even interesting (though more elementary) in the case $k = 0$, where it recovers the basic degree theory that you learn in your first course on topology. This was originally developed by Pontryagin to compute homotopy groups of spheres, but now it is regarded as a tool in manifold theory. These matters are discussed in Chapter 3 of Luck's book on Surgery theory, for example.
The second application is to physics. Unfortunately I don't understand this story very well at all, so I'll begin with what I more or less DO understand (which may or may not be well-known). The basic idea begins with the problem of situating electromagnetism in a quantum mechanical framework. Dirac began this process by imagining a "magnetic monopole", i.e. a particle that would play the role for magnetic fields that the electron plays for electric fields. The physical laws for a charged particle sitting in the field determined by a magnetic monopole turn out to depend on a choice of vector potential for the field (the choice is necessarily local), and Dirac found that changing the vector potential corresponds to multiplying the wave function $\psi$ for the particle by a complex number of modulus 1 (i.e. an element of U(1)). If we think of the magnetic monopole as sitting at the origin, then these phases can naturally be regarded as elements of a principal $U(1)$-bundle over $M = \mathbb{R}^3 - \{0\}$. But $M$ is homotopy equivalent to $S^2$, and principal $U(1)$-bundles over $S^2$ are classified by $\pi_1(U(1)) = \mathbb{Z}$. Proof: think about the Hopf fibration. The appearance of the integers here corresponds exactly to the observation of Dirac (the Dirac quantization condition) that the existence of a magnetic monopole implies the quantization of electric charge. It is remarkable to note that Hopf's paper on the Hopf fibration and Dirac's paper on magnetic monopoles were published in the same year, though neither had any clue that the two ideas were related!
The story goes on. The so-called "Yang-Mills Instantons" correspond in a similar way to principal $SU(2)$ bundles over $S^4$, which are classified by $\pi_3(SU(2)) = \mathbb{Z}$. Again, the integers have important physical significance. So these two classical examples motivate the computation of $\pi_1(S^1)$ and $\pi_3(S^3)$, but as is always the case this is just the tip of an iceberg. I am not familiar with anything deeper than the tip, but I have it on good authority that physicists have become interested in homotopy groups of other spheres as well, presumably to classify other principal bundles (it seems like a bit of a coincidence that the groups which came up in these examples are spheres, but maybe one reduces homotopy theory for other spaces to homotopy theory for spheres). People who know more about physics and/or the classification of principal bundles should feel free to chime in.
A great reference for the mathematician who wants to learn something about the physics that I discussed here is the book "Topology, Geometry, and Gauge Fields: Foundations" by Naber.
Here are some answers on the HHSvKT - I have been persuaded by a referee that we ought also to honour Seifert.
These theorems are about homotopy invariants of structured spaces, more particularly filtered spaces or n-cubes of spaces. For example the first theorem of this type
Brown, R. and Higgins, P.~J. On the connection between the second relative homotopy
groups of some related spaces. Proc. London Math. Soc. (3) \textbf{36}~(2) (1978) 193--212.
said that the fundamental crossed module functor from pairs of pointed spaces to crossed modules preserves certain colimits. This allows some calculations of homotopy 2-types and then you need further work to compute the 1st and 2nd homotopy group; of course these two homotopy groups are pale shadows of the 2-type.
As example calculations I mention
R. Brown ``Coproducts of crossed $P$-modules: applications to second homotopy groups and to the homology of groups'', {\em Topology} 23 (1984) 337-345.
(with C.D.WENSLEY), `Computation and homotopical applications of induced crossed modules', J. Symbolic Computation 35 (2003) 59-72.
In the second paper some computational group theory is used to compute the 2-type, and so 2nd homotopy groups as modules, for some mapping cones of maps $ Bf: BG \to BH$ where $f:G \to H$ is a morphism of groups.
For applications of the work with Loday I refer you for example to the bibliography on the nonabelian tensor product
http://groupoids.org.uk/nonabtens.html
which has 144 items (Dec. 2015: the topic has been taken up by group theorists, because of the relation to commutators) and also
Ellis, G.~J. and Mikhailov, R. A colimit of classifying spaces.
{Advances in Math.} (2010) arXiv: [math.GR] 0804.3581v1 1--16.
So in the tensor product work, we determine $\pi_3 S(K(G,1))$ as the kernel of a morphism $\kappa: G \otimes G \to G$ (the commutator morphism!). In fact we compute the 3-type of $SK(G,1)$ so you can work out the Whitehead product $\pi_2 \times \pi_2 \to \pi_3$ and composition with the Hopf map $\pi_2 \to \pi_3$.
These theorems have connectivity conditions which means they are restricted in their applications, and don not solve all problems! There is still some interest in computing homotopy types of some complexes which cannot otherwise be computed. It is also of interest that the calculations are generally nonabelian ones.
So the aim is to make some aspects of higher homotopy theory more like the theory of the fundamental group(oid); this is why I coined the term `higher dimensional group theory' as indicating new structures underlying homotopy theory.
Even the 2-dimensional theorem on crossed modules seems little known or referred to! The proof is not so hard, but requires the notion of the homotopy double groupoid of a pair of pointed spaces. See also some recent presentations available on my preprint page.
Further comment: 11:12 24 Sept.
The HHSvKT's have two roles. One is to allow some calculations and understanding not previously possible. People concentrate on the homotopy groups of spheres but what about the homotopy types of more general complexes? One aim is to give another weapon in the armory of algebraic topology.
The crossed complex work applies nicely to filtered spaces. The new book (pdf downloadable) gives an account of algebraic topology on the border between homotopy and homology without using singular or simplicial homology, and allows for some calculations for example of homotopy classes of maps in the non simply connected case. It gets some homotopy groups as modules over the fundamental group.
I like the fact that the Relative Hurewicz Theorem is a consequence of a HHSvKT, and this suggested a triadic Hurewicz Theorem which is one consequence of the work with Loday. Another is determination of the critical group in the Barratt-Whitehead n-ad connectivity theorem - to get the result needs the apparatus of cat^n-groups and crossed n-cubes of groups (Ellis/Steiner).
The hope (expectation?) is also that these techniques will allow new developments in related fields - see for example work of Faria Martins and Picken in differential geometry.
Developments in algebraic topology have had over the decades wide implications, eventually in algebraic number theory. People could start by trying to understand and apply the 2-dim HHSvKT!
Edit Jan 11, 2014 Further to my last point, consider my answer on excision for relative $\pi_2$: https://math.stackexchange.com/questions/617018/failure-of-excision-for-pi-2/621723#621723
See also the relevance to the Blakers-Massey Theorem on this nlab link.
December 28, 2015 I mention also a presentation at CT2015 Aveiro on A philosophy of modelling and computing homotopy types. Note that homotopy groups are but a pale "shadow on a wall" of a homotopy type. Note also that homotopy groups are defined only for a space with base point, i.e. a space with some structure. My work with Higgins and with Loday involves spaces with much more structure; that with Loday involves $n$-cubes of pointed spaces. As with any method, it is important to be aware of what it does, and what it does not, do. One aspect is that the work with Loday deals with nonabelian algebraic models, and obtains, when it applies, precise colimit results in higher homotopy. One inspiration was a 1949 Theorem of JHC Whitehead in "Combinatorial Homotopy II" on free crossed modules.
Best Answer
A few comments on applications that aren't covered by the above Wikipedia article.
I don't know any applications to cryptography. Most cryptosystems require some kind of one-way lossless function and it's not clear how to do that with the complexity of the homotopy groups of spheres. Moreover, the homotopy-groups of spheres have a lot of redundancy, there are many patterns.
There's work by Fred Cohen, Jie Wu and John Berrick's where they relate Brunnian braid groups to the homotopy-groups of the 2-sphere. It's not clear if that has any cryptosystem potential but it's an interesting aspect of how the homotopy-groups of a sphere appear in a natural way in what might otherwise appear to be a completely disjoint subject.
Homotopy groups of spheres and orthogonal groups appear in a natural way in Haefliger's work on the group structure (group operation given by connect sum) on the isotopy-classes of smooth embeddings $S^j \to S^n$. I suppose that shouldn't be seen as a surprise though. Moreover, it's not clear to me that this is always the most efficient way of computing these groups. But I think all techniques that I know of ultimately would require some input in the form of computations of some relatively simple homotopy groups of spheres.
I think one of the most natural applications of homotopy groups of spheres, Stiefel manifolds and orthogonal groups would be obstruction-theoretic constructions. Things like Whitney classes, Stiefel-Whitney classes and general obstructions to sections of bundles. Not so much the construction of the individual classes, more just the understanding of the general method.