Solve Phil's Canine Conundrum

 is a side quest in the original Fallout.

Walkthrough
Go to the Casino area of Junktown. To the right of the entrance to the area there are a pair of people trying unsuccessfully to enter their house. The reason: a dog.

Simply wear a Leather Jacket (which will remind it of it's former master) or give the pooch some Iguana-on-a-stick to solve the quest and let the couple enter their house unmolested. There is a 100 experience reward for doing this, as well as the couple's thanks and a pet dog named Dogmeat that will follow you everywhere and has lots of nifty bite attacks in combat.

Reward

 * 100 XP.
 * Dogmeat as a companion.

Behind the scenes
The whole ordeal with the dog is a reference to the movies Mad Max and A Boy and His Dog. The name Dogmeat is used for the tititular boy's dog, while the leather jacket-wearing fool is a nod to Mel Gibson's role as Max Rockatansky.