We report the dynamical behaviors of a magnetic vortex driven by an out-of-plane spin-polarized current in a point-contact geometry using a micromagnetic simulation method. Polarity, chirality, and polarity plus chirality-switching diagrams are presented corresponding to different current states. Besides vortex core gyrotropic motion, polarity switching, chirality switching, and polarity plus chirality switching, we also observe non-switching and partial switching behaviors in some cases. Investigations of these switching behaviors find that polarity is switched by nucleation and annihilation of the vortex–antivortex pair for the polarity-switching only regime, and chirality reversal is realized by moving, nucleation, and annihilation of vortices in the disk for lower current density, but by propagating of spin waves for higher current density. Moreover, the reversal of chirality is always accompanied by polarity switching, and the accompanied polarity switching for the higher current density case is realized by the expansion and compression of the vortex core, not the traditional vortex–antivortex pair-mediated mechanism.