Non-orthogonal multiple access (NOMA) has recently received much attention as a candidate technique for the fifth generation networks. In this paper, considering both the direct and relay-aid paths, we investigate the performance of a downlink NOMA-based cooperative system and further analyze two different user ordering schemes. The outage probability, diversity gain, and ergodic rate are studied as three benchmarks to evaluate the system performance. For different user ordering schemes, the exact outage probabilities of users are first solved in the closed form. Then, the outage behavior in the high signal-to-noise ratio (SNR) region is discussed to obtain the diversity gain. In addition, the closed-form expression of ergodic rate for the strongest user and upper bounds for the rest users at high SNR is provided. Finally, numerical results verify the accuracy of our analysis and demonstrate that sorting users based on the relay-aided path can provide a larger ergodic sum rate in some cases. By contrast, sorting users based on the direct path can provide a better diversity gain, and the corresponding performance is less sensitive to relay's location.