Gaussian curvature

The product of principal curvatures measures intrinsic surface curvature.

Different ways to understand it

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Geodesics
Geodesics have zero covariant acceleration and locally extremize length under suitable conditions.
Tangent spaces
Tangent vectors at a point form a linear space describing infinitesimal directions.
Arc length
Integrating speed measures a sufficiently regular curve's length.
Covariant derivatives
A connection differentiates vector fields while accounting for changing tangent spaces.
Curvature and torsion
Curvature measures tangent turning; torsion describes departure from a plane in space.