How many stages of sleep do humans have? and which stage involves constant eye motion?

Humans have 5 stages ofor sleep, stages 1,2,3,4,5 . They don't really have specific names , but for example stages 1 and 2 are typically characterized as "light sleep" Stages 3 and 4 are "Middle sleep" and part of 4 and 5 are REM. Part two of your question is REM (Rapid Eye Movement) which occurs in stages 4-5.

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Humans have 5 stages of sleep. Stage 1, Stage 2, Stage 3, Stage 4 and REM sleep, where
- Stage 1 is only a few minutes, with some sudden body movements possible
- Stage 2 your muscles get more relaxed and breathing and heart rate gets slower
- Stage 3 you will be into delta wave (which are wavelengths with low frequency, but large amplitude)
- Stage 4 activities in your brain decrease, and delta wave dominates EEG pattern 
- REM sleep your eyeballs move rapidly back and forth under the closed eyelids (hence, REM is rapid eye movement). Physiological arousal increases to similarly like daytime levels. Heart rate quickens and breathing becomes more irregular. Brain-wave activity resembles that of active wakefulness. 

So to answer your question, there are 5 stages of sleep: Stage 1, Stage 2, Stage 3, Stage 4, REM (not called stage 5). REM sleep does not include stage 4, REM sleep is a stage itself, and the stage that involves constant eye motion.