Why do we care about quotient maps? In topology class, I often hear statements like "It turns out that $f$ is always a quotient map."  I don't understand what is so special about quotients that the question of whether a given surjective map is or isn't a quotient is so interesting.  What motivates these statements, and what should I think about when someone mentions that $f$ is a quotient?
 A: Quotient maps preserve more properties of spaces than mere continuous maps do. E.g. we know that if $f:X \to Y$ is surjective and continuous and $X$ is connected then so is $Y$. The same does not hold for locally connected $X$ in general, but it does hold when $f$ has the stronger property of being a quotient map. 
It also helps that other classes of maps (open surjective maps, closed surjective maps and perfect maps) are special cases of quotient maps. 
Some properties are chacterised by quotient maps: A space is a $k$-space iff it is a quotient image of a locally compact Hausdorff space and a space is sequential iff it is the quotient image of a metrisable space, e.g. It implies that both these properties are preserved by quotient maps as well. 
Another handy fact: if $f: X \to Y$ is quotient we can test the continuity of a function $g: Y \to Z$ by seeing whether $g \circ f: X \to Z$ is continuous. This universal property of quotients is important in category theory.
A: Quotient maps are fun once you get the hang of them.
A quotient space is a partition of a space S into sets,
each of which is considered to be a point in the quotient
space.  The quotient map f, is a map from each point x, of S
to the partition part in which x resides.  
The topology of the quotient space is created by requiring
f to be continuous.  
In group theory, a simular partioning occures and the
quotient is required to be a group.  That requires a
normal subgroup to be the identity of the quotient group.  
They also can be lots of fun giving us,
for example, the integers modulus n.  
