A pseudodifferential operator is a generalization of a partial differential operator to fractional orders. Pseudodifferential operators allow you in some cases to invert a differential operator. For elliptic boundary value problems, they provide the easiest means to show that the surprising fact that the solution is smoother in the interior than on the boundary.

Here are some introductions to the subject:

• Essentials of pseudodifferential operators by Gabriel Nagy
• Some lecture notes on distributions and pseudodifferential operators by Gerd Grubb. The page is in Dutch, but the lecture notes are in English.
• Lectures on Pseudo-differential Operators by M. S. Joshi