What is a quotient ring and cosets? So I am trying to understand what a coset is and what a quotient ring is. 
So I am going to tell you guys what I know. And please let me know if my thinking is right or wrong, and if I am missing something. For the rest of this post, assume $R$ is a ring and $I$ is an ideal of that ring. 
So $I = (m)$ is a principal ideal generated by $m$ where $m \in R$.
Now the congruence class of $a$ and $I$ is denoted by $[a]_m$ but this congruence class can also be written as $a + (m)$ or simply $a + I$. 
now this congruence class is obviously a set. So is this the coset of it? So for any $a$ that you choose in $R$ and you "add" the ideal to it (generated by m which is also in $R$) you get a coset. 
The quotient ring $R/I$ just means ALL the cosets of $I$ in $R.$ So does this say that if hypothetically speaking $m = 3,$ then $1 + (3)$ is one coset, $2 + (3)$ is another coset and hence the quotient ring is ALL the cosets for every possible $a?$ 
I hope I make sense. If someone could send me a link to an easy (introduction to algebra) article or a "tutorial", that would be appreciated. I am using Hungerford Algebra. 
If anyone can explain this to me in easy english, that is appreciated. 
Thanks
 A: Remember that every equivalence relation induces a partition on the set on which you have defined the relation. An ideal $I$ defines an equivalence relation on the set $R$ by saying that $a\sim b$ if and only if $a-b\in I$; we express this by saying that $a$ is congruent to $b$ modulo $I$. 
That means that $R$ is partitioned into equivalence class under this "congruent to modulo $I$" relation. The equivalence classes are called "cosets" (I claimed some time ago this is short for "congruence set", but have been unable to substantiate this; but you can surely imagine that it is). The cosets are the equivalence classes.
Now, what is the equivalence class of an $a\in R$? it is the set of all things that are congruent to $a$ modulo $I$; this consists exactly of all elements of the form $a+x$ with $x\in I$, so we write it as
$$ a + I = \{ a+x \mid x\in I\}.$$
This is its description as a set. If we want to think of it in terms of the equivalence relation and remember that it is the equivalence class of $a$, then we use the standard notation for equivalence classes and write $[a]$ (or $[a]_I$, or $[a]_m$, to remind us also of which ideal $I$ we are dealing with). 
When you ask if $a+I$ is "the coset of it", it is unclear to me who "it" is. But, $a+I$ is the equivalence class of $a$, so it is the coset of $a$ (since "the coset of $x$" just means "the equivalence class of $x$ under the equivalence relation 'congruent modulo $I$'"). 
The reason that when you "add" the ideal you get the coset is just because of what the definition of the equivalence relation is: every element of the form $a+i$ is congruent to $a$ modulo $I$, because $a-(a+i) = -i\in I$; and if $b$ is congruent to $a$ modulo $I$, then $a-b=i$ for some $i\in I$, so we get that $b=a-i$. That is, every element of the form $a+x$ with $x\in I$ is in the coset, and everything in the coset is of the form $a+x$ with $x\in I$. So the notation $a+I$ is both suggestive and useful.
Now, the set $R/I$ is just the set of equivalence classes; as a set, the elements are the cosets. Each coset has many different names, since $[a]_I = [b]_I$ whenever $a\sim b$. 
As a ring, $R/I$ is the ring whose elements are the cosets/equivalence classes, and whose operations $\oplus$ and $\odot$ are defined by
$$\begin{align*}
[x]_I \oplus [y]_I &= [x+y]_I,\\\
[x]_I\odot[y]_I &= [x\cdot y]_I
\end{align*}$$
where $+$ and $\cdot$ are the operations in $R$. (We later drop the distinction between $+$ and $\oplus$, but the point here is that they are operations defined on different sets, so they are really different functions, though very closely related). 
In your example, yes: $R/I$ is the collection of all cosets, but remember that the same coset may have many different names. So, for instance, if $R=\mathbb{Z}$ and $I=(3)$, then $R/I$ consists of all cosets, which are sets of the form
$$ a + (3) = \{ \ldots, a+3(-2), a+3(-1) , a+3(0), a+3(1), a+3(2), a+3(4),\ldots\}$$
for all $a$. But as it happens, every single coset is equal to either $0+(3)$, $1+(3)$, or $2+(3)$, so in fact $R/I$ has only three elements, even though each of those elements has infinitely many names:
\begin{align*}
0+(3) &= 3+(3) = 6+(3) = 9+(3) =\cdots = 3k+(3),\\
1+(3) &= 4+(3) = 7+(3) = 10+(3) = \cdots = 3k+1 + (3)\\
2+(3) &= 5+(3) = 8+(3) = 11+(3) = \cdots = 3k+2 + (3)
\end{align*}
