This paper investigates the optimal power allocation scheme for sum throughput maximization of non-orthogonal multiple access (NOMA) system with $\alpha$-fairness. In contrast to the existing fairness NOMA models, $\alpha$-fairness can only utilize a single scalar to achieve different user fairness levels. Two different channel state information at the transmitter (CSIT) assumptions are considered, namely, statistical and perfect CSIT. For statistical CSIT, fixed target data rates are predefined, and the power allocation problem is solved for sum throughput maximization with $\alpha$-fairness, through characterizing several properties of the optimal power allocation solution. For perfect CSIT, the optimal power allocation is determined to maximize the instantaneous sum rate with $\alpha$-fairness, where user rates are adapted according to the instantaneous channel state information (CSI). In particular, a simple alternate optimization (AO) algorithm is proposed, which is demonstrated to yield the optimal solution. Numerical results reveal that, at the same fairness level, NOMA significantly outperforms the conventional orthogonal multiple access (MA) for both the scenarios with statistical and perfect CSIT.