What is a quotient structure?
It's a term coming from abstract algebra. When you quotient a set $S$ with respect to some equivalence relation $\sim$, you are replacing the set $S$ with its equivalence classes under $\sim$. Often the set $S$ has operations defined on it, and in that case we want these operations to be well defined for the quotient $S/\sim$, in the sense that if $x_i \sim y_i$ then $x_1 \circ x_2 \sim y_1 \circ y_2$. This allows us to define $\circ$ on the quotient.

An example in real analysis is $L^p$ spaces, in which functions are defined "up to measure zero".