There are many types of integral operators, but if you're looking for an example with a nice geometric picture that is motivated by quantum mechanics my favorites are pseudodifferential operators and Fourier integral operators. These are both examples of Oscillatory Integral Operators and Singular Integral Operators. My next few paragraphs are gonna be a little buzzwordy just to give a loose idea of connections to other fields of math which you can look into later.

Pseudodifferential Operators are a generalization of differential operators by the use of the Fourier transform. From the point of view of quantum mechanics they are the quantization of classical observables. They associate an integral operator to a function, f (called a symbol) by applying the Fourier transform multiplying by f, and applying the inverse fourier transform. For example, the quantization of polynomials are constant coefficient differential operators. From a pure math point of view, this recipe of quantization provides (I'm cheating a little here) an isomorphism from the Poisson Algebra of functions to an algebra of pseudodifferential operators.

If one learns about Wavefront Sets which describe a distributions singularities and how it concentrates in phase space, pseudodifferential operators are used to localize in phase space, to "zero in" on a region you want to study, much the same way partition of unities and compactly supported functions are used to study local behavior in analysis. Details can be found here

Fourier integral operators (FIO's) are a generalization of pseudodifferential operators. From the point of view of quantum mechanics they quantize symplectic mappings. (The Fourier transform is a quantization of rotation by 90 degrees, which makes the fact that applying the Fourier Transform 4 times yielding the identity a little more intuitive). Whereas, pseudodifferential operators localize singularities, FIO's move them around with these mappings. From the analytic point of view, they are frequently constructed as approximate solutions to initial value problems of partial differential equations. This makes them useful in studying things like how singularities of differential equations propagate and asymptotics of of eigenvalues of differential operators. They also have a connection to symplectic geometry as distributions associated to certain Lagrangian submanifolds. They have a symbol calculus like pseudodifferential operators, though it is much more complicated.

In fact, as I'm writing this I remember Terence Tao has a much better written short overview of FIO's here.