Quotient spaces until recently gave me a great deal of trouble. I had no intuition for them, and no image to latch on to.
Until I started taking algebraic topology.
I learned that the quotient under an equivalence relation is like gluing 2 spaces together, where the points you stick together are identified with each other.
This excellent image illustrates the idea for the attaching map of two topological spaces. The requirement that be a homeomorphism is also clearer from the picture.