In physics we often define the Chern number as the closed integral over the Berry curvature $$\Omega_{xy}=\frac{\partial A_x}{\partial k_y}-\frac{\partial A_x}{\partial k_x}.$$ With, $$A_i(\mathbf{k})=i\langle \psi_{\mathbf{k}}|\partial_{k_i}|\psi_{\mathbf{k}}\rangle.$$ So, we are often interested in, $$C=\frac{1}{2\pi}\oint_{}\Omega_{xy} d^2k,$$ where the Chern number is a closed integral over all momentum in the Brillouin zone.
My question is: What is a Chern number generally? Is it simply the integral over a closed surface such that the result is invariant in certain regions of the parameter space? I often see meantions of Chern classes, how are they related to Chern numbers?
If M is an closed oriented $2n$-manifold, a Chern number of $M$ is simply the integral of any product of Chern classes of its tangent bundle which has total degree $2n$.