Intelligent reflecting surface (IRS) is a promising new paradigm to achieve high spectral and energy efficiency for future wireless networks by reconfiguring the wireless signal propagation via passive reflection. To reap the potential gains of IRS, channel state information (CSI) is essential, whereas channel estimation errors are inevitable in practice due to limited channel training resources. In this paper, in order to optimize the performance of IRS-aided communications with imperfect CSI, we propose to jointly design the active transmit precoding at the access point (AP) and passive reflection coefficients of IRS, each consisting of not only the conventional phase shift and also the newly exploited amplitude variation. First, the user's achievable rate is derived assuming a practical IRS channel estimation method, which shows that the interference due to CSI errors is intricately related to the AP transmit precoder, the channel training power and the IRS reflection coefficients during both channel training and data transmission. Next, by combining the benefits of the penalty method, Dinkelbach method and block successive upper-bound minimization (BSUM) method, a new penalized Dinkelbach-BSUM algorithm is proposed to optimize the IRS reflection coefficients for maximizing the achievable data transmission rate subjected to CSI errors. Finally, simulation results are presented to validate the effectiveness of our proposed algorithm as compared to benchmark schemes. In particular, useful insights are drawn to characterize the effect of IRS reflection amplitude control (with/without the conventional phase-shift control) on the system performance under imperfect CSI.