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.

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

1. Suppose that $L$ is the tautological line bundle on the complex projective space plane, then it's clearly not trivial, and neither is $V= L\oplus L^{-1}$. But   $c_1(V)=0$, because the trace of curvature is zero for an induced connection. Of course, $V$ is certainly 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.