The significance of the puppy episode in Chapter 2 is not revealed until later in the book. It provides the basis for the clue that confirms for George, Myrtle's husband, that his wife is having an affair. Myrtle asks Tom to buy her the dog, and she keeps it at the apartment where she and Tom have their illicit meetings. George doesn't know anything about the puppy, and, although he has his suspicions, he does not actually know for sure that his wife is cheating on him either.
In Chapter 8, after Myrtle has been killed, George finds a new dog collar wrapped in tissue among Myrtle's things. The item is incongruous, because he and Myrtle do not have a dog. George concludes that Myrtle was indeed being unfaithful to him, and that she was killed by the man with whom she was having an affair. This sets in motion the chain of events that leads to the murder of Gatsby.