Mumps is an acute viral disease of humans and really only humans. It is spread through the respiratory tract (probably droplets and aerosols) where the virus most likely infects the cells lining the epithelium - we don't really know this for certain. From here it spreads throughout the body, infecting organs and tissues such as the salivary glands, testes and ovaries, pancreas, and perhaps most worryingly, your central nervous system. In fact, in the pre-vaccine era, mumps was the biggest cause of aseptic meningitis across the developed world; it can also cause the more severe encephalitis.
Lucky we've got a vaccine, right?