The Dark Energy Spectroscopic Instrument (DESI) is an ongoing spectroscopic survey to measure the dark energy equation of state to unprecedented precision. We describe the DESI Sky Continuum Monitor System, which tracks the night sky brightness as part of a system that dynamically adjusts the spectroscopic exposure time to produce more uniform data quality and to maximize observing efficiency. The DESI dynamic exposure time calculator (ETC) will combine sky brightness measurements from the Sky Monitor with data from the guider system to calculate the exposure time to achieve uniform signal-to-noise ratio (SNR) in the spectra under various observing conditions. The DESI design includes 20 sky fibers, and these are split between two identical Sky Monitor units to provide redundancy. Each Sky Monitor unit uses an SBIG STXL-6303e CCD camera and supports an eight-position filter wheel. Both units have been completed and delivered to the Mayall Telescope at the Kitt Peak National Observatory. Commissioning results show that the Sky Monitor delivers the required performance necessary for the ETC.