Oscar is Junior's dog. He is "only an adopted stray mutt, but he was the only living thing that [Junior] could depend on" (9). Oscar is actually not a main character of the story. He is introduced, sadly, to illustrate a bigger point about poverty.

Junior explains that the worst thing about being poor is not actually hunger. He then goes on to tell the story of Oscar, his adopted and beloved dog, who was "better than any human [he] had ever known" (9). But then Oscar gets sick and there's not enough money to take him to the vet. Instead of treating the sick dog, Junior's dad gets out his gun and puts him down. 

For Junior, the painful memory of Oscar and his fate is an illustration of the real physical pain of poverty. In a sense, Oscar is a piece of the bigger picture compelling Junior to leave the reservation and do something more with his life.

In The Absolutely True Diary of a Part-time Indian, who is Oscar?

In The Absolutely True Diary of a Part-time Indian, Oscar is the Spirit family's dog.  Junior characterizes the dog as being his best friend, and Junior says that besides Rowdy, he does not really have any human friends, so Oscar is incredibly important to him.  But the dog eventually gets sick, and the Spirit family cannot afford to take him to the vet.  Junior pushes his parents to do something for the dog, but they just do not have the money to care for Oscar.  So his father euthanizes the dog instead.  Junior's recount of what happens to Oscar is significant because it underscores the overwhelming poverty that his family experiences, yet it also shows how sympathetic his family is to times of hardship.  Junior's family makes hard choices out of love and concern.