Chern number from Berry curvature
Integrating Berry curvature over a closed momentum-space surface yields an integer topological invariant.
$$C=\frac{1}{2\pi}\int_{\mathrm{BZ}}\Omega_{xy}(k)\,d^2k\in\mathbb Z$$
What this means
Integrating Berry curvature over a closed momentum-space surface yields an integer topological invariant.
Deep dive: mathematical and physical mechanism
The curvature is the field strength of the Berry connection. A nonzero integral means the band bundle cannot be globally trivialized. For a two-band model $H=\mathbf d(k)\cdot\boldsymbol\sigma$, $C$ counts the signed wrapping of the Bloch sphere by $\hat{\mathbf d}(k)$, with band/sign conventions specified.
How it appears in quantum phenomena
Integer Chern numbers control Hall conductance of fully occupied isolated bands.
Assumptions, caveats and limitations
Requires a global bulk gap for the occupied subspace; individual band Chern numbers need isolated bands.
Connect the ideas
For context, see the Topological Band Theory topic guide and the reading references.