"Sleep" (verb) 

1.   To be in a resting state when the body is relaxed and the mind is unconscious.

• "I couldn't sleep even though I was so tired."

• "We slept quite late last night after the party was over."

2.   To have enough space or beds for a specific number of people to sleep for the night. 

Usually we would apply this to different types of accommodation like tents and houses, etc.

• "The studio room sleeps four if you pull out the sofa bed."

• "The tent has enough room to sleep six."

"Sleep" (noun)

1.   The resting state when the body is relaxed and the mind is unconscious.

 Quite often this is used as part of a common phrase, "get some sleep."

• "I'm exhausted. I'm going to get some sleep."

• "I finally got some sleep after we finished the project."

Idiomatic phrases with "sleep":

"Get/go to sleep" - to succeed in sleeping.

• "She always goes to sleep right after a bedtime story."

• "It took me a few hours to get to sleep last night."

"On the verge of sleep" - to come very close to going to sleep.

• "I was on the verge of sleep when I heard a loud noise."

2.   A period of sleeping. 

Quite often this is used as part of idiomatic expressions.

• "Why don't you have a little sleep before the long drive home?"

• "I fell into a deep sleep after watching a documentary last night."

• "Did you have a good night's sleep?"

3.   (informal) A way of saying how many nights away something is. 

However, this is not that common.

• "Two more sleeps until our honeymoon!"

"Sleeping" (noun) - the state or condition of being in a state of sleep.

"Sleeping" (adjective) - in or relating to the state of sleep.

• "I enjoy sleeping too much to wake up early on the weekends."

• "The sleeping baby lay peacefully in her crib."

"Asleep" (adjective) - sleeping or not awake.

• "The kids were already asleep when I got home."

Common idiomatic phrases with "asleep":

"Fall asleep" - to start to sleep.

• "They fell asleep as soon as the movie started."

"Sound/fast asleep" - completely asleep.

• "He was sound asleep for the entire journey."

"Half asleep" - not completely asleep.

• "I'm usually half asleep until I have my cup of coffee."

"Rest" (verb/noun) has a number of different definitions but it usually means to stop being active in order to relax to recover strength.

This is not the same as "sleep" because you do not have to be unconscious to rest. However "sleep" and "nap" are considered "rest," so it is occasionally used as synonyms and implies a recovery of strength or energy.

• "You'll feel better after a good night's rest."

• "She's resting in the bedroom."

"Shuteye" (noun, informal) - sleep. 

Although this is considered old-fashioned, I still hear this used from time to time.

• "I'm looking forward to getting some shuteye when I get home."

"Slumber" (verb, noun) - sleep 

This is usually used in literature and is not that common in everyday conversation.

• "She woke up from a long slumber and found herself in a new world."

"Catch/cop/get some z's" (American informal expression) - to sleep. 

This is not that common in Australia, but I sometimes hear this on American TV shows.

• "I'm going to catch some z's in the lounge before my next shift."

"Beauty sleep" (idiom) - used in a humorous way to refer to the sleep required to feel and look healthy and attractive.

• "I'm going to need my beauty sleep if I want to look good for tomorrow's presentation."

"Go to bed" - to go to the bed or place where you sleep.

This is often used to mean that one is going to sleep, but it is also used to mean that one physically moves to the bedroom to relax to eventually fall asleep - maybe to read a book, or watch TV.

• "It's been a long day. I'm going to bed."

"Turn in" (phrasal verb, informal) - to go to bed.

• "He usually turns in at about 10pm after watching his favourite TV show."

"Retire" (verb) has a number of different definitions but it can also mean to go to bed. 

However this is considered more formal and kind of old-fashioned.

• "She retired early for her big day tomorrow."

"Hit the sack/hay" (informal American expression) - to go to bed. 

This is well-known but not that commonly used.

• "I'm going to hit the sack before I end up binging the entire show."