An affectionate and dextrous puppy, Dogmeat's skills as a four-legged scavenger and ferocious fighting companion were not lost on his first master, who was recently killed by raiders in the vicinity of the Capital Wasteland scrapyard. Dogmeat has tracked the raiders down, loyal to the end.
His name is an homage to the Dogmeat from Fallout, and, according to the developers, he is a direct descendant of the Vault Dweller's original canine companion. 
Dogmeat is found in the Scrapyard (south of Minefield), fighting several raiders. Regardless of any other companion or Karma level, Dogmeat can be recruited without incident. If the event to encounter him is triggered and the Lone Wanderer does not approach him, he may wander and be found elsewhere on the map.
Shooting Dogmeat in the initial encounter without talking to him will make it impossible to recruit him unless the Puppies! perk was taken.
The following is based on Fallout 3 cut content and has not been confirmed by canon sources.
Before the release of Fallout 3, the process of how to recruit Dogmeat had only been shown as a small clip as the Lone Wanderer holds something out to the wounded dog. The item is not revealed. Dogmeat was said to be found in a random encounter, which would appear in different places depending on where the player ventures to, but was meant to appear early on to help develop the character's bond with the dog. This was scrapped in the released game.
End of information based on Fallout 3 cut content.
Dogmeat is modeled after the dog in Mad Max 2 (aka The Road Warrior). The main character Max, played by Mel Gibson, had a dog that wandered with him. Dogmeat is the same breed as Max's dog, an Australian cattle dog.
The namesake for "Dogmeat" is Vic's canine companion, "Blood" (also called "Dogmeat" by Vic) in the movie A Boy and His Dog (1975) starring Don Johnson and based on the novel by the famed sci-fi author Harlan Ellison. This movie dealt with post-nuclear war survival with "Dogmeat" apparently conversing with Vic as they both scour the wasteland for food and women.
pcps3xbox360 In Little Lamplight, if Dogmeat is following you and you go into the testing chamber to explore, Dogmeat may appear dead while exiting. Then while leaving a room or house, Dogmeat's dead body will spawn outside next to where you entered. Dogmeat may or may not take up a character slot after dying this way. Dogmeat will also still be shown blinking. This also can happen to Jericho. [verified]
xbox360 During the Blood Ties quest Dogmeat may disappear from the game after talking with Vance about making a deal with Arefu. Loading a previous save doesn't fix the problem. [verified]
ps3 The game might crash if you try to talk to Dogmeat while the battle is still ongoing in the Scrapyard. [verified]
ps3xbox360 Through use of the Paralyzing Palm perk (or any weapon with a knockdown affect), Dogmeat can store items whose weight total exceeds 10,000 lbs. However, asking him to fetch you an item at any time will return all stored items. If you ask Dogmeat to return to Vault 101, you can pickpocket him and store items too. [verified]
ps3 Despite being sent back to Vault 101, Dogmeat has shown up inside of the West residence in Arefu. After going outside, Dogmeat does not follow. [verified]
pc Attacking Dogmeat until he becomes hostile and then putting away your weapons will result in Dogmeat walking back to Vault 101. [verified]
xbox360 After leaving Vault 101 for the first time, Dogmeat can be found outside, sometimes clipping into the ground. If interacted with, no dialog options will appear and will not be recruitable as a companion, even if the player goes to the scrapyard. [verified]
pcxbox360 If the player character kills an enemy while Dogmeat is attacking them, he will not notice that the enemy is already dead and keep attacking the corpse even past it being mutilated. [verified]
ps3 Dogmeat can be brought to Point Lookout by having him search for an item just before sleeping on the cot in the Duchess Gambit. He will approach the player in Point Lookout after six days. He can be returned to the Capital Wasteland simply by taking the Duchess back, resulting in him automatically waiting at Vault 101.[verified]