What is Central force; central conservative force?

A force is central with center $O$ if it is always directed along $\vect{OM}$: $\vect F = F(M)\,\vect e_r$. It is central conservative if its magnitude depends only on $r = OM$: $\vect F = F(r)\,\vect e_r$, which derives from a potential energy $E_p(r)$ with $F(r) = -\dd E_p/\dd r$.