What does the maximal ideal of rings means? Though I know the meaning of ideal, I just couldnot figure out What does the maximal ideal of rings means? If possible please explain me the meaning of quotient ring as well.
A maximal ideal can be thought of as the maximal element of the partially ordered set $\displaystyle \left(\left\{I:I\subseteq R\text{ is an ideal}\right\},\subseteq\right)$. It's really just an ideal $\displaystyle J$ such that if $\displaystyle J'$ is an ideal with $\displaystyle J\subsetneq J'$ then $\displaystyle J'=R$. It's the 'biggest' ideal.
A quotient ring is just like all the other quotient structures in algebra. Given a two-sided ideal $\displaystyle I$ we can partition $\displaystyle R$ into equivalence classes where the equivalence relation is $\displaystyle a\sim b\Leftrightarrow a-b\in I$ and we usually denote $\displaystyle [a]$ by the coset notation $\displaystyle a+I$. We can then define a ring structure on $\displaystyle \left\{r+I:r\in R\right\}$ by $\displaystyle (r+I)+(r'+I)=(r+r')+I$ and $\displaystyle (r+I)(r'+I)=(rr')+ I$.
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.
Note that the existence of maximal ideals in an infinite ring relies on the Axiom of Choice (see Zorn's Lemma).
That is to say, assuming the Axiom of Choice, maximal ideals always exist. If you do not assume it, they may not.