What does Smoothness Prior mean in the context of image processing?
A prior is something that you know a priori (beforehand) about how the system you're looking at works.

A smoothness prior means that you know that the image you're looking at will be smooth in a particular way.

That paper goes on to say:


  The term diffusion comes from the idea of propagating local information with smoothness constraints, by analogy with physical phenomena like heat propagation in physical structures. These phenomena can be formalized with PDEs, and diffusion is therefore performed using PDE-based regularization.


So in this case the smoothness that you know about beforehand is enforced using a partial differential equation.