Why the term and the concept of quotient group?
Often I think of a quotient group in terms of (loss of) information. When we move from a group to its quotient group we lose some information about the identity of the elements. For example, when we map an element of the additive group of integers $\mathbf{Z}$ to the quotient group $\mathbf{Z}/10\mathbf{Z}$ we lose the information of all the other digits save the least significant one. In other words after moving down to the quotient group we can no longer tell the difference between 9, 999, or 314159. In this sense we then equate 9 with 99 et cetera
Why would we want do this, as it amounts to loss of information? Well, there are several reasons. Sometimes we are only really interested in the residual information. For example, when we study the set of numbers of the form $a+b\root 3\of 2+c\root 3\of 4$, where $a,b,c$ are integers, and we want to start adding, subtracting and multiplying them, we quickly notice that those operations are very similar to the corresponding operations involving polynomials $a+bx+cx^2$. The difference is that we are only interested in the value of the polynomial at a single point $x=\root 3\of 2$. This shows in the multiplication rule, because the polynomial $x^3$ takes the value $2$. In order to make this correspondence between polynomials and numbers more accurate we are forced to equate the polynomial $x^3-2$ with the polynomial $0$. This time we get a quotient ring instead of a quotient group (see algebra textbooks for such details), but the idea is that some things we have learned about polynomial algebra will carry over to our set of numbers, and that gives us the benefit of economy of thinking. We don't need to relearn everything from scratch, if the next time we are interested in $\root 3\of 3$ instead.
Sometimes quotient groups are forced upon us. We are not in possession of all the information. A simple example is the following. Assume that somebody is counting coins, but the only counting aid available to him is a light switch. Every time he tallies one more coin he will toggle the light switch: lit, dark, lit, dark,... He may or may not be able to keep track of the actual tally, but if somebody else comes to the room, or the tallyman gets confused, the status of the light switch will only tell whether an odd or an even number of coins have been counted, i.e. we have moved from the group $\mathbf{Z}$ to the quotient group $\mathbf{Z}/2\mathbf{Z}$. Another very common quotient group in mathematics is used to decribe an angle of rotation. Let's say that we are studying a planar object spinning about its center of mass. It may have completed God knows how many full revolutions, but when we enter the room and observe its position, we have no way of knowing anything else but the current direction pointed at by, say, a small arrow somebody painted on the object for this purpose. A full revolution corresponds to an angle of rotation $2\pi$, so the total angle of rotation will have an uncertainty that can be any integer multiple of $2\pi$. In other words, we can only see an element of the quotient group $\mathbf{R}/2\pi\mathbf{Z}$, not an element of $\mathbf{R}$.
Quotient groups $A/B$ count cosets of $B$ inside $A$. The counting even works well with addition.
The Cartesian plane forms a group A, and a line through the origin is a subgroup B. The cosets of B inside A are all the parallel lines. How many are there?
Suppose B is the line: $$ B = \{ (x,y) : y = 2x \}$$ or just B is y = 2x for short. Parallel lines are parameterised by their "intercept" b, the coset b + B is the line with slope 2 and intercept b. $$b+B = \{(x,y) : y = 2x+b\}$$ This means that there is exactly one coset for every real number. In some sense we have counted the parallel lines.
The way we count them even keeps track of addition. If I took a point on the line B, say (2,4), and added it to a point on 7 + B, say (3,13), then I get the point (5,17), which is on the line $$(0 + B) + (7 + B) = 7 + B.$$ If I add the point (3,7) on 1 + B to the point (8,20) on 4 + B, then I get the point (11,27) on $$(1+B) + (4+B) = 5 + B$$ This is just because if x = 11, then 2x is 22, and 27 is 2x+5.
If one wanted to be more precise, I suppose one should say (0,5) + B, since it should be an element of $A$ plus $B$, but just like $A/B \cong \mathbb{R}$, we only need one number here too.
In a sense, the quotient group is indeed a measurement of how many copies of your normal subgroup are within the larger group. In the simple example of $\mathbb{Z}/3\mathbb{Z}$, the group has three elements: one for the subgroup $3\mathbb{Z}$ itself and one for each of its two cosets, which, if you were to plot them on a number line, for example, "look" the same as the original subgroup. And if you put the subgroup and its two cosets together, you get the whole group $\mathbb{Z}$. So in a sense the quotienting in this case tells you how many subsets resembling $3\mathbb{Z}$ are needed to break down $\mathbb{Z}$. The thing that makes this different from arithmetic division, of course, is the fact that the quotient is also a group--the group structure just comes from the way the "copies" of the subgroup interact with each other.