Active reconfigurable intelligent surfaces (RISs) overcome the severe multiplicative fading effect observed in conventional passive RIS-assisted systems by amplifying the reflected signal. This paper investigates the performance of an active RIS-assisted downlink NOMA network. Closed-form expressions of the outage probability (OP) and ergodic capacity are derived in terms of the Meijer-G function. Regarding the former, we formulate the asymptotic OP and obtain the diversity order of the system. Additionally, we explore higher-order modulation schemes such as rectangular quadrature amplitude modulation (QAM), cross QAM, and hexagonal QAM, which are integral for achieving increased data rates while maintaining power and bandwidth efficiency. For these modulation schemes, we derive, closed-form expression of the average symbol error rate using the Gaussian Chebeyshev quadrature approximation. The impact of the number of reflecting elements (REs) and their amplification gain, thermal noise, and other system parameters are investigated. The performance of the considered NOMA network is also compared with an orthogonal multiple access technique. We show that the diversity order of the considered network scales with the number of REs and fading severity parameters, and the hexagonal QAM outperforms several other QAM schemes. Finally, simulation results affirm the accuracy of the derived closed-form expressions.