Definition of "quotient set"
A quotient set is what you get when you "divide" a set $A$ by $B\subseteq A$, wherein you set all elements of $B$ to the identity in $A$. For example, if $A=\Bbb Z$ and $B=\{5n\mid n\in\Bbb Z\}$, then you're making all multiples of $5$ zero for all intents and purposes, so the quotient is $\{0,1,2,3,4\}$.
Another (and more correct) way of saying this is that a quotient set is all equivalence classes on the set $A$ under a given equivalence relation. In the example above, $aRb\iff 5|(a-b)$, so clearly the equivalence classes are $n\equiv 0,1,2,3,4\pmod 5$. In reality, you can select any number from each equivalence class, so $\{20,-34,77,63,-1\}$ would be a "correct" quotient set, just not canonical.
I'm way late to the party, but for anyone stumbling upon this like I did, a simple definition for a quotient set is "the set of all equivalence classes of a set under a given equivalence relation." An equivalence class IS the same as a partition, defined by using some equivalence relation. But the quotient is ALL of those equivalence classes (partitions) under that particular equivalence relation. You DO need an equivalence relation to build a quotient set, which is why the notation is S/~, which is read as "the quotient set of the set S under the equivalence relation ~."
At the risk of over-simplifying it, you could say that the quotient set under a particular equivalence relation is the same as the original set, but in partitions rather than all together. (This isn't strictly true, but it's a useful way to understand the basic idea)
For example, say the set $C$ is the set of all cars, and $\sim_c$ is an equivalence relation that means "is the same color as." So for some white car $w$ and some other white car $h$, $w\sim_ch$. With that kind of equivalence relation, $[w]$ is the equivalence class that means "all white cars," and is a partition of the set of all cars $C$. $[h]$ works just as well since both $w$ and $h$ are in the same equivalence class. (You could write $[w]_{\sim_c}$ to be specific about which relation you're using for the equivalence class) The quotient set $C/\sim_c$ would be the set of all equivalence classes in $C$ under $\sim_c$. I.e., it's the set of all partitions of $C$, partitioned by color. If you happen to be fully color blind so that all cars look either white, grey, or black, then the quotient set $C/\sim_c$ would be $\{[w],[g],[b]\}$ (given that $g$ and $b$ are grey and black cars just like $w$ is a white car). So to build a quotient set, you either list out all of the possible equivalence classes OR generalize it with set notation: $$C/\sim_c\ \ = \{[x]_{\sim_c}\mid x\in C\}$$
Since each equivalence class in this context represents all cars with a specific color and the quotient set contains all color groups (by definition), then the quotient set ultimately still represents the same group of objects. If you were to visualize this, then you'd see all the cars in the world gathered into different groups by color rather than all in one heap. More strictly speaking, the quotient set is more like a listing of the groups rather than a listing of all the cars in the groups (After all, the elements in $C$ are cars, but the elements in $C/\sim_C$ are sets of cars). Writing $[w]$ is like saying "all cars with the same color as this $w$ one," or "this car's color is what I mean by 'white.'" (the first one is more mathematically correct, but the second makes more contextual sense with the example)
I hope that helps to thoroughly visualize the concept. In general, a quotient set differs from a partition in that a quotient set contains partitions as elements, and those partitions are defined by an equivalence relation. The two are definitely distinct.