Resting

Resting is used to heal in a safe place without the need to use stimpaks or healing skills. The player is unable to rest if there are enemies nearby, is being exposed to radiation, or if the bed is owned by someone else.

Fallout and Fallout 2
In Fallout and Fallout 2, you can rest to recover your health in almost any place, as long as you're not in combat, using the Pip-Boy 2000 alarm clock function. A handful of places, such as the toxic caves or Vault City downtown, prohibit resting.

Fallout 3
Sleeping in a bed for any length of time will restore full health and heal all limbs, but will not get rid of rad buildup. You can not sleep in an owned bed (if the owner dies the bed won't revert to an unowned state). Many places have unowned beds for players to use freely.

Alternatively, it is possible to wait in a certain location for however long you desire. Unlike sleeping, there is no benefit to this other than advancing the clock. It will not restore any health, and all the drawbacks still apply. It mainly serves the purpose of moving forward in time to a point where certain events will trigger, such as the open hours of a shop.

Well Rested
If you sleep in a bed you own or rent (such as in your Megaton house, Tenpenny Tower Suite, Kenny's Cave, or a rented room) for at least 8 hours, you will become "Well Rested" and gain a 10% bonus to experience gained for 12 hours. The add-on Broken Steel adds the perk Deep Sleep, which gives you the bonus from sleeping in any bed (provided the rest of the conditions are met). EDIT: The Well-Rested benefit does not require 8 hours of sleep and can be achieved by sleeping for a single hour in an owned/rented bed.

Bed Locations

 * The article Fallout 3 player housing provides numerous locations with "free beds".

Hotels
Sleeping in these will provide you with the "Well Rested" bonus.
 * Megaton, Moriarty's Saloon, Nova, 120 caps.
 * Rivet City, The Weatherly Hotel, 120 caps.
 * Underworld, Carol's Place, 120 caps.

Add-ons
The Point Lookout add-on provides an additional well rested bed in Kenny's Cave, provided you pass a speech check, have the Child at Heart perk, or retrieve Kenny's special teddy bear.

Fallout New Vegas
Beds act the same as in Fallout 3, but with some minor differences. In addition to owned beds conveying a Well Rested bonus, having sex with someone will also convey the bonus. In Hardcore mode, the healing benefits of a bed only apply to owned beds; other beds in the Wasteland will reduce fatigue but not restore any health.

Camp Forlorn Hope

 * In the medical tent; any unused bed can be used.

Goodsprings

 * The bed in the gas station is available, if Ringo isn't already sleeping there.
 * The bed in Victor's shack (the Cowboy robot) is also available.
 * Easy Pete's house has two beds.

Mojave Outpost

 * Unoccupied bunk beds in the Barracks are available.

Freeside

 * The King's School of Impersonation has spare beds.

Novac

 * Manny Vargas has an extra bed in his room.

Misc

 * The cots in the Nevada Highway Patrol Station (between Primm and Mojave Outpost) are available.

Sloan

 * In the Miners' Barracks.

Ownable beds

 * The Dino Dee-Lite hotel room in Novac can be owned by the player. It will be given to you permanently for free after completing One for My Baby properly, or can be purchased from Jeannie May Crawford for 100 caps. Also, if "Come Fly With Me" is completed your reputation will increase in Novac and Jennie May will offer the room for free when the 'I'd lke to rent a room' dialog is selected.
 * The Tops, Ultra-Luxe, Atomic Wrangler and the Lucky 38 casinos all have beds that can be owned, either given as quest rewards or won through gambling. The Vault 21 hotel has a rentable room that can be made free through repeated donations of Vault suits to the owner.