TITLE: Cokernels - how to explain or get a good intuition of what they are or might be
QUESTION [50 upvotes]: 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].

REPLY [5 votes]: What I am thinking, to try to get a handle on this, is that the image of a kernel vanishes - so in the "target" of some map/function/arrow one can actually recover no information about the kernel. The kernel is actually the biggest piece of the "object" of which this is true. 
Dually, if one is looking at the "image" of a function, one can say nothing about the structure of the cokernel (in terms of the part of the "target" which is not in the "image"), and this is the biggest piece of the target for which this is true. 
The distinction between the two in elementary work seems to be that one is more interested in the structure of the object, of which the kernel is a natural part.
This became too long for a comment, so I have posted it as an answer. Direct sums/orthogonal sums of vector spaces seem to provide the earliest examples of cokernels in my own mathematical education. Other examples which would be pedagogically helpful would be very much appreciated.