What do the curvature and torsion measure? Consider a smooth surface for simplicity. What does its curvature measure? What does its Gaussian/Riemannian curvature measure? What does its torsion measure?
What does the Ricci curvature measure?

The curvature literally measure how sharp a curve is at a certain point. For example if you have a circle of small radius then the curvature (which is given by $\frac{1}{R}$ for a circle) at any given point will have a large curvature which means that it is bending sharply. Similarly, if you have a large radius then the curvature will be much smaller which means that it is bending less sharply. The Torsion of a curve measure how sharply it is twisting out of its plane curvature. I perfect example to look at would be the helix. (To be specific when I talk about a small radius I am referring to $R<1$.)
The Ricci curvature tensor provides a way to which the geometry provided by the Riemannian metric might differ from ordinary Euclidean n-space. An application of the Ricci curvature tensor is in General Relativity in connection with the Einstein Field Equations. 
