The following sufficient condition for operators on $\mathbb{R}^n$ is well-known: If the kernel of $a(x,D)$ has compact support and $sup_{x} \mathop{sup}_{x} |a(x,xi)| a(x,\xi)| \to 0$ as $\xi \to \infty$ Then infty$, then A extends to a compact operator on $L^2$.
A possible reference is M. Shubin, Pseudodifferential operators and spectral theory, Second Edition, Springer.
It follows easily that on a compact manifold a classical pseudodifferential operator is compact if and only if it has negative order.
