the Ricci scalar

If the Ricci tensor is curvature averaged over directions, the Ricci scalar takes one final step and boils all of it down to a single number at each point. It is the most compressed statement of how curved a space is right here: one figure that says, on balance, whether a small ball has more or less volume than a flat-space ball of the same radius. Positive scalar curvature means space closes up on itself like a sphere; negative means it flares out like a saddle.

It is the full contraction of the Ricci tensor with the inverse metric, R = g^ab R_ab, and it carries dimensions of one over length squared. For a two-dimensional surface it is simply twice the Gaussian curvature, so for a sphere of radius a it equals 2/a^2, a positive constant. More concretely, the volume of a small geodesic ball of radius r differs from its flat-space value by a term proportional to R, so a positive Ricci scalar means neighbours are packed slightly closer than Euclidean geometry would predict.

The Ricci scalar is the star of the Einstein-Hilbert action, S = (c^4 / 16 pi G) integral of R times sqrt(-g) d^4x, whose stationary points are precisely Einstein's field equations, general relativity derived from a single principle of least action. The honest caveat is that a single scalar cannot capture the full geometry: R can be zero in a badly curved spacetime. The Schwarzschild black hole has R = 0 everywhere outside the mass, and yet it bends light and tears objects apart with tides.