Defect structures in tungsten diselenide (WSe₂) are drawing increasing attention because they play a role in the optical response and in some cases give rise to single-photon emission, providing new platforms for the application in quantum information processing^1-4. To date, however, the understanding of the nature and electronic structure of such structural defects in WSe₂ atomic layers is still lacking⁵. Here we describe the characterization of individual point defects in WSe₂ atomic layers using scanning tunneling microscopy and spectroscopy. In bilayer WSe₂ on graphene, we observe in-gap electronic states, and spatially dependent spectroscopy and dI/dV maps showed that the charge state of the defect can be manipulated. We also characterize other types of point defects in our measurements, including ones characterized by a very small energy bandgap (similar to that of a metallic phase of WSe₂⁶) and others which feature sharp electronic states near the valence band and a negative differential resistance (NDR), similar to that observed in single quantum states^7,8. Finally, we propose a minimal single-electron transport model explaining these effects.