Reconfigurable intelligent surface (RIS) has emerged as a cost- and energy-efficient solution to enhance the wireless communication capacity. However, recent studies show that a very large surface is required for a RIS-assisted communication system; otherwise, they may be outperformed by the conventional relay. Furthermore, the performance gain of a RIS can be considerably degraded by hardware impairments such as limited-resolution phase shifters. To overcome those challenges, we propose a novel concept of hybrid relay-reflecting intelligent surface (HR-RIS), in which a single or few elements are deployed with power amplifiers (PAs) to serve as active relays, while the remaining elements only reflect the incident signals. Two architectures are proposed, including the fixed and dynamic HR-RIS. Their coefficient matrices are obtained based on alternating optimization (AO) and power allocation strategies, which enable understanding the fundamental performances of RIS and relaying-based systems with a trade-off between the two. The simulation results show that a significant improvement in both the spectral efficiency (SE) and energy efficiency (EE) with respect to the conventional RIS-aided system can be attained by the proposed schemes, especially, by the dynamic HR-RIS. In particular, the favorable design and deployment of the HR-RIS are analytically derived and numerically justified.