This work introduces downlink non-orthogonal multiple access (NOMA) into short-packet communications. NOMA has great potential to improve fairness and spectral efficiency with respect to orthogonal multiple access (OMA) for low-latency downlink transmission, thus making it attractive for the emerging Internet of Things. We consider a two-user downlink NOMA system with finite blocklength constraints, in which the transmission rates and power allocation are optimized. To this end, we investigate the trade-off among the transmission rate, decoding error probability, and the transmission latency measured in blocklength. Then, a one-dimensional search algorithm is proposed to resolve the challenges mainly due to the achievable rate affected by the finite blocklength and the unguaranteed successive interference cancellation. We also analyze the performance of OMA as a benchmark to fully demonstrate the benefit of NOMA. Our simulation results show that NOMA significantly outperforms OMA in terms of achieving a higher effective throughput subject to the same finite blocklength constraint, or incurring a lower latency to achieve the same effective throughput target. Interestingly, we further find that with the finite blocklength, the advantage of NOMA relative to OMA is more prominent when the effective throughput targets at the two users become more comparable.