There are two types of sleep: NREM (non-rapid eye movement) sleep and REM (rapid eye movement) sleep.

During REM sleep, our eyes are shut, of course. However, underneath our eyelids, our eyes are making these strange movements. They primarily shift left and right, and occasionally they’ll move up and down.

Aserinsky, E., & Kleitman, N. (1953). Regularly occurring periods of eye motility, and concomitant phenomena, during sleep. Science, 118, 273-274. doi:http://dx.doi.org/10.1126/science.118.3062.273