The nature of intrinsic point defects, i.e., energetic, ferroelectric, and electronic properties of vacancies in ferroelectric PbTiO3, is studied using first-principles calculations based on the hybrid Hartree-Fock density functional, which correctly reproduces the band gap and thus provides the accurate defect electronic states. The oxygen vacancies are found to behave as double shallow donors and are thermodynamically stable over a wide range of Fermi levels under oxygen-poor conditions, while Pb vacancies are likely to be formed under oxygen-rich conditions and act as double shallow acceptors, which lead to p-type conductivity. The result shows good consistency with the relevant experimental observations. The ferroelectric distortion is disturbed anisotropically outward by oxygen vacancies and is characterized by localized defect electronic states contributed by d orbitals. In contrast, an isotropically inward ferroelectric disturbance is found around Pb and Ti vacancies. Such a ferroelectric disturbance is strongly enhanced by the charging of vacancies, except for Pb vacancies that induce delocalized defect states.