I have some confusion about the properties regarding quotient rings. If we have a ring, $\frac{R}{I}$ then I understand that the quotient ring is the set of all equivalence classes $[f]$ such that $f$ is in $R$. I also understand that in this case, equivalence means that two elements, $x,y$ are equivalent if $x-y \in I$. But I don't quite understand what the set of all these equivalence classes means. Does this mean all sets of points in $R$ such that $x-y \in I$? Some concrete examples or other suggestions would be very helpful in understanding this, thank you.
Also, my confusion is specifically for when $R$ is a ring of polynomials.