Index theory is fundamentally about a homomorphism
$$K_n(M) \to \mathbb{Z}$$
from the top degree K-homology of $M$ (even dimensional) to the integers called the *analytic index map*. It is called this because every graded self-adjoint elliptic (pseudo)differential operator on $M$ determines a class in $K_n(M)$ and the analytic index map sends the class of an elliptic operator to its index. The content of the index theorem is that the analytic index map agrees with another map (the *topological index*) from K-homology to the integers defined via algebraic topology (essentially the Poincare-Thom construction).

Baum and Douglas constructed a geometric model of the K-homology of a CW-complex $X$ in which a cycle consists of a spin$^c$ manifold equipped with its spin$^c$-Dirac operator and a reference map from the manifold into $X$. They proved that the abelian group generated by these cycles (with relations that I will not specify) really is the K-homology of $X$, i.e. the homology theory determined by the Bott spectrum. The classical index theorem for spin$^c$-Dirac operators is really about calculating the topological index of the fundamental class of a spin$^c$-manifold.

So since the Fredholm index is well-defined on K-homology classes (the existence of the analytic index map) and K-homology classes can be represented by Dirac operators (Baum-Douglas), this means that any index problem can be reduced to an index problem involving Dirac operators. There are index theorems involving operators which are not obviously related to Dirac type operators - Toeplitz operators come to mind (this is related to Sonke Hansen's answer), for instance - but even for those operators the index theory is controlled by Dirac operators.

Now some comments about the role of pseudodifferential operators in all this. Pseudodifferential operators typically enter into index theory for one of three reasons:

- To prove that a certain operator is Fredholm.
- To construct homotopies.
- To emphasize the stability of the index.

First, to prove that an operator is Fredholm it is typically easiest to show that it is invertible up to compact operators. One of the goals of pseudodifferential operator theory is to construct pseudo-inverses for differential operators in this sense, and many people like to prove that differential operators are Fredholm by explicitly constructing pseudo-inverses. This is more or less the line that Atiyah-Singer took in their original papers.

Second, to do calculations in index theory one often must construct homotopies between different operators (exploiting the homotopy invariance of the Fredholm index), and even if you only care about differential operators these homotopies usually must pass through pseudodifferential operators.

Finally, the K-homology picture of index theory emphasizes how insensitive the Fredholm index is to the details of a differential operator. When constructing the K-homology class of an elliptic operator (using the Baum-Douglas model or otherwise) one notices that the construction depends only on the asymptotic properties of the principal symbol; for example, if the principal symbols of two different operators agree outside a compact set at each point then the K-homology classes will be the same. Thus the index theorem for pseudodifferential operators comes almost for free, and I think this is why index theorems are often formulated for pseudodifferential operators (rather than to handle a specific set of examples).

Dirac operator D of Sis the first order differential operator given by the following composition $$C^\infty(S) \to C^\infty(T^\ast M \otimes S) \to C^\infty(TM \otimes S) \to C^\infty(S),$$ where the first arrow is given by applying the connection, the second by identifying $T^\ast M$ and $TM$ via the Riemannian metric of M, and the third arrow is given by the Clifford action. An operator of Dirac-type is one which arrises in that way. Especially, the symbol of D is given by Clifford multiplication: $\sigma_\xi(D)s = i\xi \cdot s$. $\endgroup$