A maximal ideal can be thought of as the maximal element of the partially ordered set . It's really just an ideal such that if is an ideal with then . It's the 'biggest' ideal.

A quotient ring is just like all the other quotient structures in algebra. Given a two-sided ideal we can partition into equivalence classes where the equivalence relation is and we usually denote by the coset notation . We can then define a ring structure on by and .

To show that's well-defined isn't obvious, and for anything more you'll need to see a book. I suggest Dummit and Foote or Herstein.