sectional curvature
The full Riemann tensor in high dimensions is a bulky object with many components, and it is hard to picture all at once. Sectional curvature is the human-scale way to read it: pick a point, pick a flat 2-dimensional slice of directions through that point, and ask how curved the space is along that particular slice. The answer is a single number — and it is exactly the Gaussian curvature of the little surface swept out by geodesics in those two directions.
Precisely, at a point p choose a 2-dimensional plane P inside the tangent space, spanned by two independent tangent vectors u and v. The sectional curvature K(P) is a specific ratio built from the Riemann tensor: K(P) = g(R(u, v)v, u) divided by (g(u, u) g(v, v) - g(u, v)^2), where the denominator is the squared area of the parallelogram on u and v, making the answer independent of which u, v you chose to span P. Geometrically, K(P) is the Gaussian curvature, at p, of the surface formed by sending out geodesics in all directions lying in P. Knowing K(P) for every 2-plane P at every point recovers the entire Riemann tensor, so sectional curvature is full information, just repackaged plane by plane.
Sectional curvature is the most geometric of the curvature notions because each value is a familiar Gaussian curvature. Its sign has vivid meaning: positive sectional curvature makes nearby geodesics converge (as on a sphere, where meridians meet at the poles), zero leaves them parallel (the flat plane), and negative makes them spread apart exponentially (hyperbolic space). Whole theorems classify manifolds by the sign or pinching of their sectional curvature. A caution: in dimension 3 or higher, sectional curvature can vary wildly from plane to plane at the same point, so a single 'curvature of the space' rarely exists — you must say in which 2-plane.
On a sphere of radius a, every 2-plane through every point gives the same sectional curvature K = 1/a^2 — the space is equally curved in all directions, which is why it is a constant-curvature model. On a saddle-shaped surface, by contrast, the curvature in the direction along the valley and the direction up the ridge have opposite signs, giving negative Gaussian curvature.
Sectional curvature is the Gaussian curvature of a 2-direction slice; on the sphere it is the same in every slice.
Sectional curvature is attached to a 2-plane of directions, not to a point alone. Above dimension two it generally differs from plane to plane, so 'the curvature here' is ambiguous until you name the plane.