There are two main types of conflict, internal conflict (man verses himself) and external conflict (man verses man, nature, and supernatural). Internal conflict tends to be the result of one's internal struggle regarding his or her morality and desires. External conflict exists as the result of one conflicting with outside forces (another person/group, natural forces (such as animals or the weather), or the supernatural (such as gods, demons, ghosts, or goblins).
As for the conflict depicted in Toni Cade Bambara's "Raymond's Run," the protagonist (Hazel Elizabeth Deborah Parker, or Squeaky) faces both internal and external conflict.
One internal conflict Squeaky faces exists is taking care of her brother ("All I have to do in life is mind my brother Raymond, which is enough."). Squeaky does not have to do housework, hustle money, or anything else. All she is responsible for is taking care of Raymond. By stating that taking care of Raymond "is enough," one can infer (make an educated guess) that she feels conflicted about it. Her exasperation is evident.
Externally, Squeaky's conflict lies in beating Gretchen in the May Day race. Gretchen is Squeaky's only competition, and Squeaky cannot lose to her. Beating Gretchen actually exists as both an internal and external conflict. Externally, Squeaky must run faster than Gretchen. Internally, Squeaky will not feel as if she is "the best" if she loses.