The future sixth generation (6G) is going to face the significant challenges of massive connections and green communication. Recently, reconfigurable intelligent surfaces (RIS) and non-orthogonal multiple access (NOMA) have been proposed as two key technologies to address the above problems. Motivated by this fact, we consider a downlink RIS-aided NOMA system, where the base station seeks to communicate with two NOMA users with the aid of a RIS. Considering future network supporting real-time service, we investigate the system performance with the view of effective capacity (EC), which is an important evaluation metric of sensitive to delay sensitive system. Based on this basis, we derive the analytical expressions of the EC of the near and far users. To obtain more useful insights, we deduce the analytical approximation expressions of the EC in the low signal-to-noise-ratio (SNR) approximation by utilizing Taylor expansion. In order to compare, we provide the results of orthogonal multiple access (OMA). It is found that 1) The number of RIS components and the transmission power of the base station have important effects on the performance of the considered system model. 2) Compared with OMA, NOMA system has higher effective capacity due to the short transmission time.