Short answers, pulled from the story.
The Riemann curvature tensor measures the failure of parallel transport to return a vector to its original direction after traveling around a closed loop on a curved manifold. It also quantifies the non-commutativity of second covariant derivatives. A manifold has zero Riemann curvature if and only if it is flat, meaning locally isometric to Euclidean space.
The tensor is named after Bernhard Riemann and Elwin Bruno Christoffel. The classical method for deriving it was developed by Gregorio Ricci and Tullio Levi-Civita, whose Ricci identity expresses the tensor as the commutator of the covariant derivative of a covector with itself. The first Bianchi identity was also discovered by Ricci.
The Riemann curvature tensor is the central mathematical tool in general relativity, the modern theory of gravity. It represents the tidal force experienced by a rigid body moving along a geodesic, in a sense made precise by the Jacobi equation. The curvature of spacetime is observable in principle via the geodesic deviation equation.
The number of independent components follows from the tensor's symmetries: two skew symmetry conditions, the algebraic Bianchi identity, and interchange symmetry. These symmetries are equivalent to the tensor belonging to the image of the Young symmetrizer corresponding to the partition 2+2. For a two-dimensional surface, the Bianchi identities reduce the tensor to a single independent component, fully determined by the Gaussian curvature.
The Ricci curvature tensor is obtained by contracting the first and third indices of the Riemann curvature tensor. On a two-dimensional surface, the Ricci curvature tensor is given by the Gaussian curvature multiplied by the metric tensor, and the Gaussian curvature equals exactly half the scalar curvature of the surface.
A Riemannian manifold is a space form if its sectional curvature is equal to a constant. The Riemann tensor of a space form has a specific form involving that constant, and conversely, except in dimension 2, if the curvature takes that form for some function, the Bianchi identities force that function to be constant, confirming the manifold is locally a space form.