sectional curvature
The full Riemann tensor is a four-index monster; sectional curvature is the friendly, geometric number you extract from it by asking a simple question. Pick a point p and a 2-dimensional plane sigma through p (spanned by two tangent vectors). Sweep out the little surface made of geodesics in those directions, and measure its Gaussian curvature at p. That number K(sigma) is the sectional curvature of the plane — curvature reduced to the Vol I picture of a surface's bending, plane by plane.
Given two linearly independent tangent vectors u, v at p spanning the plane sigma, K(sigma) = g(R(u,v)v, u) / ( g(u,u) g(v,v) - g(u,v)^2 ). The denominator is the squared area of the parallelogram on u, v, which makes K independent of the basis chosen for sigma — it depends only on the plane. Geometrically K(sigma) equals the Gaussian curvature at p of the 2-dimensional surface exp_p(sigma) swept out by geodesics tangent to sigma. The key structural fact: knowing K(sigma) for every 2-plane sigma at every point recovers the entire Riemann curvature tensor — the sectional curvatures contain exactly the same information as R.
Where it lives and an honest caveat. Constant sectional curvature characterizes the space forms (sphere, Euclidean, hyperbolic). Sectional curvature is the FINEST of the curvature scalars: from it you can build Ricci by averaging over planes, and scalar by averaging again, but you cannot go back — a Ricci or scalar bound is strictly weaker and does not determine the sectional curvatures. So 'positively curved' must specify which curvature: positive sectional curvature is a far stronger hypothesis than positive Ricci or positive scalar curvature, and the three classes of theorems are genuinely different.
The round n-sphere of radius a has constant sectional curvature K = 1/a^2 for every plane at every point; Euclidean R^n has K = 0; and hyperbolic n-space has constant K = -1/a^2. These three constant-curvature cases are precisely the model space forms, and a small geodesic triangle has angle sum greater than, equal to, or less than pi according to the sign of K.
Constant sectional curvature +, 0, - gives the sphere, flat space, and hyperbolic space; triangle angle sums follow the sign.
Sectional, Ricci, and scalar curvature carry strictly decreasing information. A positive-Ricci or positive-scalar hypothesis does NOT recover sectional curvature, so theorems about each are genuinely different — never silently upgrade one bound to another.