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Can anybody define what we mean by curvature of a vertex?

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Angular defect.

In modern terms, the defect at a vertex [...] is precisely the curvature at that point [...], as established by the Gauss–Bonnet theorem.

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It is perhaps worthwhile noting that this quote is referring to the integrated curvature "concentrated" at the vertex. The Gauss–Bonnet theorem relates the Euler characteristic to the total curvature integrated over the entire manifold; in generalizing this to polyhedral surfaces, the curvature would be given by a sum of delta distributions located at the vertices, with coefficients given by the angular defects of the vertices, so that integrating over the curvature would yield the sum of the angular defects. –  joriki Sep 30 '12 at 14:56
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