What is the geometrical interpretation of Ricci tensor?
The local geometric structure of a pseudo-Riemannian manifiold $M$ is completely described by the Riemann tensor $R_{abcd}$. The local structure of a manifold is affected by two possible sources
Matter sources in $M$: The matter distribution on a manifold is described by the stress tensor $T_{ab}$. By Einstein's equations, this can be related to the Ricci tensor (which is the trace of the Riemann tensor = $R_{ab} = R^c{}_{acb}$. $$ R_{ab} = 8 \pi G \left( T_{ab} + \frac{g_{ab} T}{2-d} \right) $$
Gravitational waves on $M$. This is described by the Weyl tensor $C_{abcd}$ which is the trace-free part of the Riemann tensor.
Thus, the local structure of $M$ is completely described by two tensors
$R_{ab}$: This is related to the matter distribution. If one includes a cosmological constant, this tensor comprises the information of both matter and curvature due to the cosmological constant.
$C_{abcd}$: This describes gravitational waves in $M$. A study of Weyl tensor is required when describing quantum gravity theories.
I've always liked the interpretation you get from the Raychaudhuri equation. It shows you that the Ricci tensor tends to cause geodesics to focus together. If you begin with a family of geodesics with tangent vector $u^a$, you can define the expansion $\theta\equiv \nabla_a u^a$ which measures the rate at which geodesics are spreading out or converging together. As you move along a an integral curve of $u^a$, the Raychaudhuri equation tells you how the expansion changes as a function of curve's parameter, $\lambda$: $$ \frac{d}{d\lambda}\theta = -\frac13\theta^2-\sigma_{ab}\sigma^{ab}+\omega_{ab}\omega^{ab}-R_{ab}u^au^b.$$ $\sigma_{ab}$ is called the shear and is related to the tendency of a cross section of the curves to distort toward and ellipsoid, and $\omega_{ab}$ is the vorticity and describes how the curves twist around each other. The Ricci tensor appears in this equation with a minus sign, so that when $R_{ab}u^au^b$ is positive, it tends to decrease the expansion, which describes focusing of the geodesics.