Featuring direct communications between two user equipments (UEs) without signal relay through a base station, 3GPP sidelink transmissions have manifested their crucial roles in the Long-Term Evolution (LTE) Advanced (LTE-A) for public safety and vehicle-to-everything (V2X) services. With this successful development in LTE-A, the evolution of sidelink transmissions continues in 3GPP New Radio (NR), which renders sidelink an inevitable component as well as downlink and uplink. Targeting at offering low latency, high reliability and high throughout V2X services for advanced driving use cases, a number of new sidelink functions not provided in the LTE-A are supported in NR, including the feedback channel, grant-free access, enhanced channel sensing procedure, and new control channel design. To fully comprehend these new functions, this paper therefore provides essential knowledge of 3GPP NR sidelink transmissions, including the physical layer structure, resource allocation mechanisms, resource sensing and selection procedures, synchronization, and quality-of-service (QoS) management. Furthermore, this paper also provides performance evaluation to assess the gains brought from the new control channel design. As NR sidelink transmissions have been regarded as a foundation to provide advanced services other than V2X in future releases (e.g., advanced relay), potential enhancements are also discussed to serve the urgent demand in corresponding normative works.