Meaning of "a mapping factors over another"? I was wondering what "a mapping factors over another mapping" generally means? Does it have something to do with commutative diagram in category theory?
I have seen this usage in different situations, and would like to give examples, but cannot recall them one by one except the most recent one from Fiber bundle:

Mappings which factor over the projection map are known as bundle
  maps.

Thanks and regards!
 A: The most general, broader description of your question is that you have two sets $A,B$ and maps $f,g$ into a common third space $C$, i.e., you have ; $f:A\rightarrow C$, and you have $g\colon B\rightarrow C$, and then you want to see if there exists a map $h$ between $A$ and $C$, so that "the diagram commutes" (if you ever try to do algebraic topology you will see this everywhere; in analysis, almost everywhere ; ) , i.e., if you follow the arrows and do the composition from $A$ to $B$ to $C$ , composing $g\circ h$, you will get the same result as if you go along the arrow from $A$ to $C$ alone. It is difficult to generalize because the results will depend on the structure of $A,B$ and $C$ given.
In the case of bundles you mention, a bundle map associated with a bunlde $\pi\colon E\rightarrow B$ is a map that sends an element $b$ in the base to its fiber under $\pi$, so that, if you compose back the fiber element, you get $b$ back, i.e., $\pi \circ f(b)=b$ e.g., if your fiber is a vector space, your bundle map will send $b$ to an element in the vector space, so that the composition with $\pi$ will project back down to $b$.    
A: As was pointed out in the comments:
Given a morphism $f:A \to C$ and a morphism $g: A \to B$ then $f$ is said to factor over $g$ if there exists $h: B \to C$ such that $hg = f$. Note that $h$ is unique as soon as $g$ is an epimorphism.
In the case of bundles $\pi_{E}: E \to M$ and $\pi_F: F \to N$ then a map $\varphi: E \to F$ is a bundle map if there exists $f:M \to N$ such that $\pi_{F} \varphi = \varphi \pi_{E}$, as is noted further down on the wikipedia page you linked to. So $\varphi : E \to F$ is a bundle map if and only if $\pi_{F} \varphi$ factors over $\pi_{E}$ via a (necessarily unique) map $f: M \to N$, as the bundle projections are assumed to be surjective. Note that this means in particular that $\varphi$ maps fibers to fibers.
Similarly, there is the notion of factoring through (you need to scroll down a little).
I'm mostly using this for the situation $f: A \to C$ and $h: B \to C$ then $f$ factors through $h$ if there is $g: A \to B$ such that $f = hg$. If $h$ happens to be a monomorphism then $g$ is unique (if it exists).
However, the distinction I'm making in this post are far from universally accepted.
You'll also find $f:A \to C$ factors through $B$ if there are maps $g: A \to B$ and $h: B \to C$ such that $f = hg$ and so on, $f$ factors through $g$, $f$ factors through $h$ in this same situation.
Basically, all it means is that $f$ can be decomposed as $f = hg$ in some way that should be obvious from the context. To repeat, if either $h$ is a monomorphism then $g$ is unique and if $g$ is an epimorphism then $h$ is unique.
