In this paper, a bifurcation tree of period-1 to period-8 motions in a nonlinear Jeffcott rotor system is obtained through the discrete mapping method. The bifurcations and stability of periodic motions on the bifurcation tree are discussed. The quasi-periodic motions on the bifurcation tree are caused by two Neimark bifurcations (NBs) of period-1 motions, one NB of period-2 motions, and four NBs of period-4 motions. The specific quasi-periodic motions are mainly based on the skeleton of the corresponding periodic motions. One stable and one unstable period-doubling (PD) bifurcations exist for the period-1, period-2, and period-4 motions. The unstable PD bifurcation is from an unstable period-m motion to an unstable period-2m motion, and the unstable period-m motion becomes stable. Such an unstable PD bifurcation is the third order source pitchfork bifurcation. Periodic motions on the bifurcation tree are simulated numerically, and the corresponding harmonic amplitudes and phases are presented for harmonic effects on periodic motions in the nonlinear Jeffcott rotor system. Such a study gives a complete picture of periodic and quasi-periodic motions in the nonlinear Jeffcott rotor system in the specific parameter range. One can follow the similar procedure to work out the other bifurcation trees in the nonlinear Jeffcott rotor systems.