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Channel estimation is one of the main challenges for implementing reconfigurable intelligent surface (RIS)-aided communication system with a large number of antennas at the base station (BS) because RIS is equipped with many passive reflective elements without active transmitting/receiving and signal processing abilities. In this paper, we focus on the channel estimation in a general RIS-aided multi-user mmWave communication system. Specifically, a novel two-phase channel estimation scheme consisting of on-line and off-line phases is proposed to estimate the BS-RIS channel, RIS-user channel, and BS-user channel, respectively. Inspired by the characteristics of few scatterers in mmWave communication systems, the antenna domain channels are converted into the virtual angular domain channels by using the discrete Fourier transform (DFT) matrix. Thus, the estimation problem can be formulated as a compressed sensing (CS) problem, and solved by using efficient vector approximate message passing (VAMP) algorithm with expectation-maximization (EM) to learn unknown parameters and obtain the estimates simultaneously. Simulation results show that, the above algorithm with two choices of prior distributions in the proposed mmWave RIS-aided multi-user system has fast convergence speed and desirable estimation performance. [ABSTRACT FROM AUTHOR]