When I think about kernels, I have many well-worked examples from group theory, rings and modules - in the earliest stages of dealing with abstract mathematical objects they seem to come up all over the place, whenever I see a homomorphism.
BUT no-one really seems to mention cokernels until you get to commutative diagrams and category theory. And then they can easily just be "things which make the diagram work" with limited intuition or sense of useful reality. [maybe I exaggerate]
So I am looking for good examples to illustrate what a cokernel is, extending to non-trivial examples [I was taught about the kernels of homomorphisms between non-abelian groups before anyone taught me about modules].