Intuition concerning Schwartz kernels of Operators Consider a (for example differential) operator $A$ acting on an appropriate function space over a smooth compact manifold without boundary. 

Using the Schwartz kernel $K(x,y)dy$ of the operator, its action can be described as
$$
Af(x) = \int_M K(x,y)f(y)dy\,.
$$

Sometimes one may also have families of operators $A_t$ depending on a parameter which may for example describe time.

I was wondering whether there are "intuitive ideas" behind the kernel $K(x,y)$, for example
1) what does its value at the point $(x,y)$ tell us about the action of the operator? 
2)How can I translate something like a discontinuity into some insight regarding the behaviour of the abstract operator? (e.g. kernels of inverses to differential operators, which are pseudodifferential operators, are discontinuous at the diagonal) 

Many thanks for your insight!
A large part of modern harmonic analysis is about inferring properties of the operator $A$ like being bounded from $L^p$ to $L^q$, from properties of the kernel $K(x,y)$ and in particular its behavior near the diagonal. See, e.g., the theory of Calderon-Zygmund operators.
Someone will likely correct me, but I wouldn't expect too much. $\{K(x,y)\}_{(x,y)}$ is the continuous version of the matrix representation of the operator. And usually you don't get much information about an operator by looking at the entries of its matrix in some basis.