Terahertz (THz) communication is a key candidate for the upcoming age of beyond-fifth-generation mobile networks (B5G) or sixth-generation mobile networks (6G) in the next decade and can achieve ultra-high data rates of dozens of gigabits or even terabits per second. As the carrier frequency increases from radio frequency (RF) to the THz band, the impact of meteorological factors on the wireless link is expected to become more pronounced. In this work, we first provide an overview of the attenuation caused by atmospheric gases, fog, and rain on terrestrial THz wireless communications using the recommendations of the International Telecommunication Union-Radiocommunication (ITU-R). Measured data from the literature are used to predict the attenuation caused by snow. Because unfavorable weather conditions may harm sensitive measurement equipment, ray-tracing (RT) simulations are sometimes used as an alternative to extend sparse empirical data. In this study, the terrestrial channel in an urban scenario at 300 GHz, with a bandwidth of 8 GHz, is characterized using RT simulations under different meteorological factors. The key performance parameters are explored, including path loss (PL), Rician K-factor (KF), root-mean-square (RMS) delay spread (DS), and four angular spreads. The channel characteristics under different meteorological conditions studied in this work are expected to aid the design of future outdoor terrestrial THz communications.