The quotient space is the set of equivlance classes. How one pictures it, if one should even bother doing so depends on the context.
Given where you've posted this, I guess you mean things like:
Consider RxR with the relation (x,y)~(u,v) iff x-u and y-v are integers.
With experience, you instantly notice that is the torus.
How? Imagine the plane. We identify firstly all the x coordinates with the same non-integer component, and as we go from 0 to 1 we 'wrap' round again to 0, so that's like rolling the plane up into a big cylinder. similiary in the y direction we wrap the cylinder into itself.
Obviously for more complicated examples we can't even picture the initial space, never mind using that to construct the quotient space in our heads.
An equivalence class is the set of all points that are equivalent under an equivalence relation. again, experience is the best thing here.
An equivalence relation is the same thing as a partition of a set.
What are the equivalence classes of some group G when the relation is x~y if there is a z such that zx=yz?
Can you show that's an equivalence class?
It'd help to know what level of material you're used to.
