Non-orthogonal multiple access (NOMA) is very promising for future wireless systems thanks to its spectral efficiency. In NOMA schemes, the effect of imperfect successive interference canceler (SIC) has dominant effect on the error performances. In addition to this imperfect SIC effect, the error performance will get worse with the channel estimation errors just as in all wireless communications systems. However, all literature has been devoted to analyze error performance of NOMA systems with the perfect channel state information (CSI) at the receivers which is very strict/unreasonable assumption. In this paper, we analyze error performance of NOMA systems with imperfect SIC and channel estimation errors, much more practical scenario. We derive exact bit error probabilities (BEPs) in closed-forms. All theoretical analysis is validated via computer simulations. Then, we discuss optimum power allocation for user fairness in terms of error performance of users and propose a novel power allocation scheme which achieves maximum user fairness.