Non-orthogonal multiple access (NOMA) is currently considered as a promising technology for the next-generation wireless networks. In this paper, the error rate performance of NOMA systems is investigated over Nakagami-m fading channels, while considering imperfect successive interference cancellation. In particular, this paper focuses on the pairwise error probability (PEP) analysis, where exact PEP expressions are derived to characterize the performance of all users under different fading conditions. The obtained PEP expressions are then used to derive an exact union bound on the bit error rate (BER). Through the derived PEP expressions, the asymptotic PEP analysis is presented to investigate the maximum achievable diversity gain of NOMA users. Moreover, using the derived BER bound, the power allocation problem for all users in NOMA systems is considered under average power and users BER constraints, which allows realizing the full potential of NOMA. Monte Carlo simulation and numerical results are presented to corroborate the derived analytical expressions and give valuable insights into the error rate performance of each user and the achievable diversity gain.