Sleep is a measurement of the raw time you've spent awake: the longer you are awake, the higher your Sleep value is.
Beverages containing a bit of caffeine will reduce this value, as will stopping your adventure and taking a nap. Furthermore, sleeping in a bed you own or have rented will get you the Well Rested bonus for 12 hours.
As your sleep deprivation worsens, your Agility and other stats will be affected. Not sleeping for two weeks will cause you to die.
The longer one goes without sleeping, the worse their sleep deprivation becomes. Sleep deprivation can be staved off by sleeping in a bed, but there are three bed types:
Sleeping bags can only be slept in for three hours at a time and will reduce sleep deprivation effects only to the level of Overtired.
Mattresses can be slept in for 5 hours at a time and clears all fatigue effects.
The player needs to sleep at least two hours for the curative effects to appear. Insomnia halves the effectiveness of sleep, requiring the player to sleep twice as much to remove the effects.
Certain beverages, including Nuka-Cola and its other varieties, are caffeinated and give one a brief grace period from sleep deprivation: 2.33 game hours with regular caffeinated drinks and 7 hours with Quantum.
PCPlayStation 3Xbox 360 In Fallout: New Vegas, sometimes the SLP meter in your Pip-Boy can get stuck on zero, and in fact will not display any number below the indication arrow. During this time your character will not advance in sleep deprivation. This bug may be fixed by finding a bed and sleeping for any amount of time. After the player wakes, the gauge will begin displaying a number again and will increase over time normally.
↑Fallout 4 loading screen hints: "In Survival, large doses of Caffeine, such as that found in Nuka Cola Quantum, gives you a short reprieve from your tiredness, possibly buying you time to finish a fight or find a bed."