Movable antenna (MA) systems have recently attracted significant attention in the field of wireless communications owing to their exceptional capability to proactively reconfigure wireless channels via flexible antenna movements. In this paper, we investigate the resource allocation design for an MA array-enhanced downlink non-orthogonal multiple access (NOMA) system, where a base station deploys multiple MAs to serve multiple single-antenna users. Our goal is to maximize the sum rate of all users by jointly optimizing the transmit beamforming, positions of MAs, successive interference cancellation (SIC) decoding order, and users' corresponding decoding indicator matrix, while adhering to constraints on the maximum transmit power and finite MA moving region. The formulated problem is inherently highly non-convex, rendering it challenging to acquire a globally optimal solution. As a compromise, we propose a low-complexity two-stage optimization algorithm to obtain an effective suboptimal solution. Specifically, in stage one, the SIC decoding order is first determined by solving a channel gain maximization problem. Then, in stage two, with the given SIC decoding order, the beamforming vectors, MA positions, and users' decoding indicator matrix are iteratively optimized by capitalizing on alternating optimization, successive convex approximation (SCA), and genetic algorithm (GA). Simulation results unveil that the sum-rate performance of the proposed MA-enabled downlink NOMA system significantly outperforms that of conventional fixed-position antenna (FPA) systems. Moreover, the results also show that the antenna position optimization in the proposed algorithm can further enhance the advantages of NOMA over space division multiple access (SDMA).