  In this paper, geometric characterizations of conformally flat and radially
flat hypersurfaces in $\mathbb{S}^n \times \mathbb{R}$ and $\mathbb{H}^n \times
\mathbb{R}$ are given by means of their extrinsic geometry. Under suitable
conditions on the shape operator, we classify conformally flat hypersurfaces in
terms of rotation hypersurfaces. In addition, a close relation between radially
flat hypersurfaces and semi-parallel hypersurfaces is established. These
results lead to geometric descriptions of hypersurfaces with special intrinsic
structures, such as Einstein metrics, Ricci solitons and hypersurfaces with
constant scalar curvature.
