A classical geometrical interpretation for the squared length of the second fundamental form, or Casorati curvature, of a surface in Euclidean three-space will be recalled and extended for submanifolds with arbitrary codimension in a Riemannian manifold. We will argue that the Casorati curvature coincides with our "intuitive" notion of curvature. This curvature is then used to obtain an optimal inequality between the intrinsic scalar curvature and an extrinsic curvature invariant of a submanifold in a Riemannian space form. The equality case in the inequality characterizes totally quasi-umbilical submanifolds.