Suppose we have a 3-d manifold $M$ and a point $p$ in $M$, and let $T,N,B\in T_pM$ be three orthonormal tangent vectors. We know that combinations like $(T,N,T,N)$ and $(T,B,T,B)$ and so on are sectional curvatures (I'm using DoCarmo's notation here), and they have a very clear geometrical meaning.
But what about expressions like $(T,N,T,B)$ and so on? Do they have a clear geometrical interpretation, possibly related to sectional or Ricci curvature?
In this situation, since dimension is $3$,
$$ R(T, N, T, B) = R(T, N, T, B) + R(N, N, N, B)+ R(B, N, B, B) = \pm \text{Rc} (N, B).$$
So $R(T, N, T, B)$ is measuring the Ricci curvature.
Of course there is nothing special about $R(T, N, T, B)$. We can show that the curvature tensor $R$ is determined by the Ricci curvature in dimension 3: see this answer.