What Exactly Are Quotient/Factor Groups and Rings? I'm having a lot of trouble wrapping my head around this one. The mathematical definition is that $G'$ is a factor group of $G$ if $G'$ = $G$/$N$, where $N$ is some normal subgroup of $G$. $R'$ is a factor ring of $R$ if $R'$  = $R$/$I$, where $I$ is an ideal.

My question is, just "what the hell" are they exactly? Are we taking all elements of $N$ (and similarly $I$) and taking them out of $G$ and $R$? If so, why? Why is that important? If not, what exactly are we doing when we create a factor group/ring? I could really use some help visualizing these. Thank you.
Another important way to think about quotient groups: they are precisely the images of group homomorphisms.

This is a consequence of the first isomorphism theorem: if $G$ and $H$ are groups and $\phi : G \to H$ is a group homomorphism, then $G/K \cong \operatorname{im}\phi$, where $K = \operatorname{ker}\phi$.

Conversely, if $N$ is any normal subgroup of $G$, then there is a natural homorphism $\phi : G \to G/N$ given by $\phi(g) = gN$. The image of this homomorphism is $G/N$, and the kernel is $N$.

Similar remarks apply to quotient rings.
The basic idea is that whenever you have a sub-group(ring), this sub-object allows you to classify the elements of the big group(ring) into different set-theoretical classes or "boxes". 

How? In the easiest way you can imagine: consider a sub-group $H\subset G$ and $a,b\in G$. We say these elements are in the same box (class) iff the sets $aH=bH=\{bh:h\in H\}$. The same is done with rings, but using the commutative structure (if $I\subset R$ and $a,b\in R$, then $a\sim b$ iff $a-b\in I$).

Now, these are just set-theoretical decompositions of your original algebraic objects, but if additionally the sub-group(ring) is normal(ideal) these boxes inherit the algebraic structure from the original group(ring)!