In the same vein as Kate and Scott's questions, why are fusion categories interesting? I know that given a "suitably nice" fusion category (which probably means adding adjectives such as "unitary," "spherical," and "pivotal"), we get a subfactor planar algebra which, in turn, gives us a subfactor. Also, I vaguely understand that these categories give us Turaev Viro TQFTs.
What else do fusion categories do? What's a good reference for the Turaev Viro stuff?
Fusion categories (over $\mathbb{C}$) are a natural generalization of finite groups and their behavior over $\mathbb{C}$. The complex representation theory of a finite group is a fusion category, but there are many others. In fact, you can think of a fusion category as a non-commutative, non-cocommutative generalization of a finite group. A finite-dimensional Hopf algebra is that too, but they don't have to be semisimple, while the semisimple ones give you many fusion categories, but again not by any means all of them.
Many of the basic results about the structure and representation theory of finite groups generalize, or seem like they could generalize, to fusion categories. This principle has been worked out to a very incomplete but interesting extent by Etingof and others. For instance there is an analogue of the theorem that the dimension of complex irrep of a finite group $G$ divides $|G|$. (Addendum: A qualified analogue, as Scott and Noah point out. If the category is braided, it is a strict analogue; otherwise it is an analogue of dividing $|G|^2$.) There are also semisimple Hopf algebras and other fusion categories that look a lot like $p$-groups.
You can think of the whole theory as a rebooted theory of finite groups. However, we are miles and miles away from any fusion category equivalent of the classification of finite simple groups. It is a struggle to make fusion categories that are not derived very closely from finite groups, or do not come from quantum groups at roots of unity. Only a few types of examples are known, and who knows what else is out there.
One enticing thing that does change is that dimensions of irreducible objects in a fusion category don't have to be integers. For instance, one of the simplest fusion categories is the Fibonacci category. It has two irreducible objects, the trivial one $I$ and the other object $F$. The dimension of $F$ is the golden ratio, as you can infer from the branching equation $F \otimes F \cong F \oplus I$. (But the dimensions are algebraic integers, and even cyclotomic algebraic integers. Hence divisibility is still a sensible question.)
You could also ask, why the semisimple case. As you learn in undergraduate or basic graduate representation theory, the semisimple representation theory of a finite group is much cleaner than the modular representation theory in positive characteristic.
And yes, you also get 3-manifold invariants and subfactors.
For references: Really Turaev and Viro's original paper, state sum invariants of 3-manifolds and 6j symbols, is pretty good. The generalization to spherical categories is due to Barrett and Westbury, Invariants of Piecewise-Linear 3-manifolds. And there is a discussion in Turaev's book.
A sketch: Recall that a basis-independent expression in tensor calculus has the structure of a graph with vertices labelled by tensors and edges labelled by vector spaces. A monoidal category allows the evaluation of similar expressions, except that the graph must be planar and acyclic. In a rigid pivotal category, there are good duals and the graph just needs to be planar. In a spherical category, left trace equals right trace, so a closed graph can be drawn on a sphere. If it is spherical, rigid, and semisimple, then you can use the graph of a tetrahedron to make a local interaction on the tetrahedra of a triangulated 3-manifold, and the result up to normalization is the Turaev-Viro 3-manifold invariant. (In this setting you should dualize the tetrahedra, so that a tensor morphism in the category is associated to a face of the tetrahedron.)
I am very far from an expert on the subject, but I have had to answer this question before. I will provide a few of my answers that seem to be omitted from the above discussion.
Fusion rings, or fusion rules, appear in certain physical "thought experiments" (though I am not aware that they have yet actually been observed). For instance "anyons" are defined by certain fusion rules,and if they existed would be useful in quantum computation. In the dogma of field theory, for these fusion rules to be physically meaningful, there needs to be a categorification of them, which would be a fusion category. This initiates a broad class of problems: given a fusion ring, decide whether or not it has a categorification, and if so, how many (up to equivalence). By Ocneanu's rigidity theorem, there are only finitely many categorifications of any given fusion ring, so it's possibly a "tame" problem to solve.
There is no hope (at present, per Greg's comment =]) to completely classify fusion categories in any sense, so far as I understand. The classification of fusion categories would encompass not only the classification of finite groups, but also of compact Lie groups (via the tilting module construction on the associated quantum group which yields a fusion category). So people classify fusion categories in small classes under the assumption that group theoretical categories (ones defined purely in terms of group theory: representations of groups, cohomology of groups, morita equivalences, etc.) is "easy" and they want to study the difference between the two contexts.
Greg mentioned that studying fusion categories is like studying semi-simple Hopf algebras, except that (a) there isn't necessarily a fiber functor to vector spaces, and (b) even if there does abstractly exist one, you don't choose one. If one admits the interest in studying Hopf algebras, then one has to admit the interest in fusion categories as a sort of "basis free" version. A direct application to finite groups is pinning down the precise relation between the groups D_8 and the quaternions. They are obviously not isomorphic; however their group rings are morita equivalent as rings (since they have the same number of irreducibles). Their irreps even have the same dimensions, so one can ask if their fusion categories are equivalent as fusion categories (they are not in this case, but there are some non-isomorphic groups which are so-called "isocategorical" meaning that not only are their group rings isomorphic, but the Hopf algebras are twist equivalent as Hopf algebras. The most sensible way to prove this sort of statement is through fusion categories.
For me, I am a fairly concrete-minded person, but someone who nevertheless tries to understand modern algebraic geometry, algebraic topology and category theory as best I can. Fusion categories have been a fantastic discovery for me, because they are in many ways homotopy theoretic/higher category-type constructions, but they are about as simple as one can get (because you basically have constrained the 1-morphisms as much as possible by the semi-simplicity assumption, and just focus on the higher morphisms). So for instance the first 2-groupoid I was ever able to understand in completely concrete terms arises in a paper of Etingof Nikshych and Ostrik about fusion categories. As such they can be viewed as a kindergarten of higher categories.
By the way, there is also some interest in "finite tensor categories" which are not semi-simple but satisfy the other finiteness conditions of fusion categories. (so you have finitely many simple objects, and you posit that every object is a finite-length extension of the simples). There's actually a great deal of the theory from fusion categories which generalizes here. So far as I can tell, the only obstacle in developing this notion more completely is that no one has had time to do it yet.
I also wrote a sequence of blog posts explaining the Turaev-Viro construction from the point of view of planar algebras. It has pretty pictures and might be relevant.
(Unitary) fusion categories are interesting in physics because they classify gapped phases on the boundary of 2+1D quantum states of matter. Similarly, unitary modular tensor categories are interesting in physics because they classify gapped phases of 2+1D quantum states of matter. I have two papers to explain the connections arXiv:1405.5858 and arXiv:1311.1784 .
The relationship between (sufficiently nice) fusion categories and subfactor planar algebras is a little complicated. There are three things to keep in mind.
Almost trivially, a pivotal fusion category gives an unshaded planar algebra. This is just unravelling definitions. This doesn't give a subfactor planar algebra, however.
Alternatively, you can take the "alternating part" of your fusion category $\mathcal{C}$, with respect to your favourite object $V$. This essentially defines a shaded planar algebra with $\mathcal{P}_k = \operatorname{End}(V \otimes V^* \otimes ... V)$ ($k$ tensor factors). This is a useful and interesting subfactor valued invariant of the pair $(\mathcal{C}, V)$.
Finally, a categorical Morita equivalence between two fusion categories is precisely a finite-depth subfactor (the two fusion categories are the A-A and B-B bimodules, the Morita equivalence and its inverse are the A-B and B-A bimodules).
(Warning, all of the above may require adding adjectives ...)
The analogy with finite groups is not quite as strong as Greg suggests. If you're looking at a braided fusion category then the dimensions of objects all divide the the global dimension (the sum of the squares of the dimensions).
The actual theorem (thanks Noah!) is: given $X$ an object of $Z(\mathcal{C})$, the double of a fusion category $\mathcal{C}$, $\operatorname{dim}(X)$ divides $|\mathcal{C}|$. When $\mathcal{C}$ is already braided, $Z(\mathcal{C})$ includes into $\mathcal{C}$, and this gives the statement above.
However, for general fusion categories this is only a conjecture, and indeed a conjecture with a proposed counterexample (we're still working on this one...)
When we were trying to understand Turaev-Viro and how the representation theory of $u_q(sl(2))$ mirrored the classical representation theory, we wrote The classical and Quantum 6j Symbols. This does not help with fusion categories per se, except that the representations of the quantum group must be a fusion category (correct me someone if I mis-stated that). But it does give the algebraic underpinning to the Kauffman Lins approach to TV invariants.
Fusion categories axiomatise fusion rules, that is how the tensor product of irreducibles breaks up into irreducible. This is important mathematically, but it is also important in physics when we model a physical system via a Hilbert space of states. This is important in QM (the same modelling can be fine for Classical Mechanics via the Koopman-von Neumann theory, and so the importance of this is not just restricted to QM).
More, a key role is played by the fusion rules of 2d CFT where the symmetry algebra is the Virasaro algebra and the reps are conformal families associated with a primary field and the tensor product is realised by OPEs.