General:
* Rest
* Slumber
* Doze
* Nap
* Snooze
* Nod off
* Catch some shut-eye
* Turn in
* Hit the sack
Specific to types of sleep:
* Deep sleep: Unconscious, profound sleep
* Light sleep: A state of sleep where you can be easily awakened
* REM sleep: Rapid eye movement sleep, characterized by vivid dreams
Figurative:
* Dormant: Inactive, as if asleep
* Hibernating: Sleeping through winter
* Lethargic: Sluggish and drowsy
* Comatose: In a deep, unconscious state
Informal:
* Hit the hay: Go to bed
* Catch forty winks: Take a short nap
* Go to sleep: The most straightforward synonym
The best synonym to use will depend on the specific context.