This is a big topic which should be covered in the union of many standard texts (Chern, Griffiths-Harris, Milnor-Stasheff...).  I'll a list few answers off the top of my head.

 1. Suppose that $L$ is tautological line bundle  on projective space) then it's clearly not trivial, and neither is $V= L\oplus L^{-1}$. But  we $c_1(V)=0$, because the trace of curvature is zero for an induced connection. Of course, $V$ is clearly not trivial. So it's natural to look for higher cohomological obstructions (as Arun suggested), e.g. $c_2(V) = -c_1(L)^2\not=0$ would work.
 2. For universal bundles on Grassmanians, Chern classes have natural geometric interpretations involving Schubert cycles.
 3. Chern classes come up in formulas expressing answers to natural geometric questions: Gauss-Bonnet, Riemann-Roch, or more general index theorems.