Relations of Characteristic classes: Chern, Stiefel-Whitney, Pontryagin, Euler, Wu class. We have a quite some characteristic classes: 

Chern class, Stiefel-Whitney class,  Pontryagin class, Euler class, Wu class, etc.

I wonder whether some math experts can use simple words and basic equations to give explanation on their relations and correspondence between them? (it is a more or less on the illuminating the concept.)
 A: Here goes.  
The classes that are assigned to real bundles are Stiefel-Whitney Classes, and Wu classes. The Wu class is a refinement of the Stiefel-Whitney class, it is sort of like a (Steenrod) square root.   Both take on values in $\mathbb{Z_2}$ cohomology.  They can be seen as obstructions to nonvanishing sections defined on subsets of the base.
If the bundle $p:E\rightarrow B$ is in addition oriented with fiber a vector space of dimension $r$, then the classes defining the orientation in $H^r(F,F-\{0\})$ can be woven together to form the Euler class. If $X\subset B$ is a cycle of dimenison $r$ it assigns to the cycle the intersection of number of a generic section over the cycle with the $0$-section.
The pontryagin classes are integral cohomology classes assigned to real vector bundles by turning the real vector bundle into a complex vector bundle so that a real basis for each fiber becomes a complex basis for the fibers of the new bundle.
Finally, the chern classes are assigned to complex vector bundles.  There are many geometric interpretations of the chern classes, but there was one that was more illuminating to me than the others when I finally learned it, it's due to Grothendieck.   You can build a complex vector bundle by covering the base $B$ with open sets $U_{\alpha}$ along with a continuous functions, $U_{\alpha}\cap U_{\beta}\rightarrow GL_r\mathbb{C}$ that act like they should be coordinate changes for trivializations.  To build the bundle you treat them as such.
So... a vector bundle is a lot like a matrix, and a matrix has a characteristic polynomial. The chern classes of a bundle are the coefficients of a "characteristic polynomial" of the vector bundle.  Specifically, if $p:E\rightarrow B$ is a complex $r$-plane bundle, let $P(E)\rightarrow B$ be the bundle whose fibers are the projective $r-1$-spaces modeled on the fibers.  There is a class $\gamma \in H^2(P(E))$ whose restriction to each fiber is the chern class of the canonical bundle over that fiber. The cohomology $H^*(P(E))$ is a free module over $H^*(B)$ under the obvious action with basis $, 1,\gamma,\gamma^2,\ldots, \gamma^{r-1}$.
Hence you can write $\gamma^r+c_{1}\gamma^{r-1}+\ldots +c_{r-1}\gamma+c_r=0$ where the $c_i\in H^{2i}(B)$. These coefficients are the chern classes of the bundle.
