The five types of conflicts are internal (character vs. self) and four external types: character vs. character, character vs. nature, character vs. society, and character vs. supernatural.
The internal conflicts are often related to the external ones. Basically, an internal conflict is a conflict between a character and his or her self. It is fear, frustration or decision-making. In this story, the narrator is not sane, so his internal conflicts are affected by that. From paragraph six:
"And have I not told you that what you mistake for madness is but over-acuteness of the sense?"
The narrator’s main internal conflict is his anxiety about whether what he has done will be found out. This is directly connected to the external character vs. character conflict he has with the police, who search the house, and with the man he kills. He says,
“It is impossible to say how first the idea entered my brain; but once conceived, it haunted me day and night.”
Character vs. society conflicts often involve laws. In this case, the narrator will be arrested, sent to jail and likely executed if it is discovered that he killed his roommate. In paragraph nine, the narrator states that,
“They heard! – they suspected! – they knew! – they were making a mockery of my horror!”
Character vs. nature is harder to define in this case. You could argue that the narrator’s disease is a character vs. nature conflict, also directly related to the internal conflicts of anxiety and paranoia caused by his illness.
Character vs. supernatural applies here in terms of the narrator feeling that his roommate had an evil eye. He actually states in the third paragraph,
“[F]or it was not the old man who vexed me, but his Evil Eye.”