Hearken! And observe how healthily- how calmly I can tell you the whole story.
One of his eyes represented that of a vulture- a pale blue eye with film over it. Whenever it fell upon me, my blood ran cold...
The narrator starts off explaining how wise and sane he is. He states that people think he's crazy but he says he's not. He wants to tell his story of how calmly he killed the old man with a crazy eye to prove that he's a rational, normal man.
The conflict in the story is an internal conflict which the narrator versus the narrator's obsession and hatred towards the old man's eye.
Every night, for seven nights the narrator comes into the room with the lantern and watches the man sleep as he's coming up with a plan for how he will kill the man.
On the eighth night, the old man awoke from his sleep because he heard someone in his room. The narrator took advantage of this and killed him that night by putting a bed on top of him and cutting off his limbs and placing his body under the floorboards.
The neighbor called the police when they heard a shrieking coming from the house. When the police came, he seemed to remain calm. But as soon as they sat down and started talking, the narrator thinks he can hear the heartbeat of the old man coming from the floorboards. The narrator assumes the police men can hear it too and are just taunting him so he confesses to his crime.
After he turned himself in, he immediately got arrested.