Ricci  tensor represents gravity in general relativity. It does not give the full curvature for dimension greater than three.

The embedding is a necessity to make the Riemann curvature more

consistent with the purpose of distinguishing a curved spacetime from the flat space of special relativity. Indeed, Riemann mentions in his historical paper that his curvature cannot distinguish between a plane or a cylinder or a cone. This problem  was left open and a solution of it was presented by Schlafli in 1871, based on the reasoning that the local curvature cannot be an absolute concept as Riemann proposed. The curvature as a notion of shape is a relative concept, which is valid only when we have a second object to compare. Therefore, Schlafli suggested the isometric embedding of a manifold into another, so that we could always compare the curvature of that manifold with the curvature of the embedding space. The isometric condition is to guarantee that the metric of the embedded manifold is induced by the background. For more information, please look at the various and detailed articles of Prof. Bang-Yen Chen.

A very good reply from the both experts earlier to me has been mentioned. I also want to seek attention of answer seeker, that Ricci tensor comes into existence after the process of contraction applied to Riemann tensor, however if one wants to recover riemann tensor from ricci one, one need  to go through the associate tensor. But we should keep in mind that while contracting or while raising the tensor, we always lose data. For more detail Tamra G. Kolda's work can be considered. 

As a small addition to the previous answers, it might be useful to know that there is an orthogonal decomposition of the Riemann curvature tensor into: the scalar curvature piece, the traceless Ricci tensor piece and the Weyl tensor piece. In dimension 4, the Weyl tensor further decomposes into a self-dual and an anti-self-dual part (assuming the manifold is oriented, say). Dimensions 2 and 3 are very special. In dimension 2, you basically only have the scalar curvature part, and in dimension 3, you only have the Ricci part (that is to say, you only have the scalar curvature part, and the traceless Ricci part). For more details, you can consult for instance Besse's Einstein manifolds, somewhere towards the beginning of the book.

Concerning the Ricci curvature and scalar curvature in relativity theory, the Ricci tensor is also related to the matter content of the universe via Einstein's field equation. It is the part of the curvature of spacetime that determines the degree to which matter will tend to converge or diverge in time.

The scalar curvature is the Lagrangian density for the Einstein-Hilbert action that yields the Einstein field equations through the principle of least action.