In this paper, a comprehensive analytical approach is presented for modeling the counting statistics of active quenching and passive quenching single-photon avalanche diode (SPAD) detectors. It is shown that, unlike ideal photon counting receiver for which the detection process is described by a Poisson arrival process, photon counts in practical SPAD receivers do not follow a Poisson distribution and are highly affected by the dead time caused by the quenching circuit. Using the concepts of renewal theory, the exact expressions for the probability distribution and moments (mean and variance) of photocounts in the presence of dead time are derived for both active quenching and passive quenching SPADs. The derived probability distributions are validated through Monte Carlo simulations and it is demonstrated that the moments match with the existing empirical models for the moments of SPAD photocounts. Furthermore, an optical communication system with on-off keying and binary pulse position modulation is considered and the bit error performance of the system for different dead time values and background count levels is evaluated.