I've been working through the heat-equation proof of the Atiyah-Singer index theorem. My question is what is the motivation for the definition of the index of an operator? I know there is the isomorphism between the homotopy classes of maps from a compact topological space to the space of Fredholm operators and the first K-group of the topological space given by the index map, but is there a simpler example of why we want to study the index?
Usually one really wants to know the dimension of the kernel of an operator rather than its index. The problem is that the dimension of the kernel is not a continuous function of the operator, so is very hard to compute in terms of topological data. The key point about the index is that it is a continuous function of the operator, which makes it a lot easier to compute. And it's a modification of the kernel, so with some luck one can compute the kernel from it.
I would say that the basic reason why the index of an operator is an interesting number is that one may care about the dimension of the vector space of smooth solutions of the corresponding differential equation. The index theorem serves to describe this number in terms of the symbol of the operator (in one formulation, using cohomology, in another, using $K$-theory). From my point of view, observations about the homotopy type of the space of Fredholm operators have a place in explaining why the theorem is true, not why it is interesting or useful.
Let me interpret the question as saying: "It's clear why one would be interested in (the dimension of) the kernel of a given differential operator, but why on earth would one be interested in the difference between the dimension of the kernel and the cokernel?"
I suppose one answer is that one can change the dimension of the (co)kernel of a Fredholm operator by adding a compact operator, whereas it is the difference (i.e., the index) which remains invariant. This makes it easier to compute the index, since we can deform the operator without changing its index to another operator for which the computation is easier.
Moreover, in many interesting cases one can show that cokernel vanishes, in which case the index agrees with the dimension of the kernel, which is what we were after all along.
One of my favourite examples of this is the computation of the dimension of the moduli space of instantons, as in the classic paper of Atiyah, Hitchin and Singer on Deformations of instantons.