In General Relativity which is based on Riemannian geometry curvature measures the energy density in space.

Geometrically, it is the change of double covariant differentiation. Faical Barzi wrote you what does it present in physics.

Curvature is the central subject in Riemannian geometry. It measures distance between a manifold and Euclidean spaces. As we know definition of flatness determine by Euclidean spaces. So we say a manifold is flat if its Riemannian curvature vanish. Also you can construct a relation between curvature and topological invariants, like Betti numbers. So we could classify the manifolds by curvature. On the other hand manifolds could be done flat  by some of useful and important transformations (conformal, concircular, projective, etc.)  and these flatness could determine by curvature tensors (projective curvature tensor, conformal curvature tensor, etc.).  

As Nenad Vesić said curvature is a measure of the non-commutativity of the covariant derivative, which is equal to the ordinary derivative in Euclidean spaces and always commutes and hence the curvature vanishes on Euclidean spaces as mentioned by Inan Unal.

It is for finding infinitesimal geometry of Riemannian manifolds with dimension greater than 2

Rughly speeking, Riemannian curvature tensor measures a manifold  failure to be flat. So I agree with Inan Unal's answer. It measures distance between a manifold and Euclidean spaces. So if a Riemannian curvature tensor vanish, then a manifold is Euclidean.

If one moves a vector along a loop in a Euclidean space parallelly, then one will arrive in the initial direction after returning to its original position. The Riemann curvature tensor directly measures the failure of this in a Riemannian manifold.

See page 275 of the book: Geometry, Topology and Physics by Mikio Nakahara.

The Riemann curvature is linked to the ordinary curvature of an immersed submanifold in an Euclidean space through the Theorema Egregium of Gauß. For example for an immersed surface, it is the product of the principal curvatures, which in turn are the inverse radius of the principal curvature lines. The curvature is most generally encoded in a tensor with four indices, the Riemann tensor, that by successive contractions gives the Ricci tensor and the scalar curvature. The Riemann tensor can be calculated from the metric, which is the effect of the Theorema Egregium, passing by the connection.

the meaning of riemann curvature is , that it is a measure for the deviation of vector, when it is parallel transported around a parallelogram, from the starting vector. this best seen, when you look at the definiton of the riemann curvature tensor as the commutator of two covariant.derivatives. a covariant derivative is nothing but an infinitesimal parallel transporter. hence the commutator represents a parallel transport of a vector around an infinitesimal parallelogram

I think curvature is a number.

If curvature is a number, does it mean that the Riemann curvature tensor is also a number?

Thank you.