TITLE: how do you visualize characteristic class?
QUESTION [26 upvotes]: For cohomology, there are some equivalent definitions when the object we consider is sufficiently nice. Since I mainly work with algebraic variety, I will restrict the objects I am considering to be algebraic variety.
There are several ways I think of cohomology namely:

singular cohomology (valuation of simplices)
de Rham cohomology (differential form)
Chow ring (closed subvariety)

I personal prefer thinking in the third way since it seems to be more geometric and easier for me to work with.
However, for characteristic classes, I find it hard to visualize. I know the technical details about them and some of their useful properties but I just don't get a feeling of it. Since I usually work over complex number, I personally more interested in how people think of Chern class.
Thank you!

REPLY [8 votes]: This is a variation on standard answers (to this standard question).  I'll start with a bit of philosophy since you mentioned different ways to think of cohomology.  Singular cohomology is fairly "transcendental" since to define a cochain one must assign a value to each and every chain.  When we teach basic topology, homology seems easier since we can draw/ conjure up finite linear combinations of such chains, at least in first examples.  
The latter two ways to define cohomology you cited make the assignments of values to chains geometrically - de Rham theory through integration, and sub manifolds through intersection.
So indeed, one can use your preferred intersection theory approach (which is mine too) to define characteristic classes as vanishing/independence loci.  But one can be slightly more direct in this setting.  First, triangulate your manifold and restrict to the simplicies, or if you wish just pull-back your bundle over a smooth chain (with some transversality assumptions so what's below works).
For the Stiefel-Whitney-Chern classes, take $n - i + 1$ sections of the original bundle.  The value of the $i$th SW or Chern class on a simplex will be the number of times these are linearly dependent over that simplex.
Schubert classes can be defined by embedding the bundle in a trivial bundle, and then counting the number of times a fiber over a simplex respects a given flag.
This Schubert style of geometry for characteristic classes is something I've also seen in my work on cohomology of symmetric groups, where Giusti, Salvatore and I give similar interpretations of characteristic classes of covering spaces.