To start this quest, the Vault Dweller needs to 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 they can not enter is that a dog blocks the door.
Simply wearing a leather jacket (which will remind the dog of its former master) or giving the pooch some iguana-on-a-stick will solve the quest and let the couple enter their house unmolested. There is a 100 XP reward for doing this, as well as the couple's thanks and a pet dog named Dogmeat as a companion, complete with lots of nifty bite attacks in combat.