The narrator himself gives the reason why he only killed the old man on the eighth night. He had been visiting his room at midnight for seven nights in a row expecting to see the old man's wasted eye open. The eye, "the eye of a vulture -- a pale blue eye with a film over it," had become an obsession with the narrator for, from the first time he had seen it, it made his blood turn cold. The eye horrified him and he decided to kill the old man and thus rid himself of having to see the horrible eye ever again, as he states,

I made up my mind to take the life of the old man, and thus rid myself of the eye forever.

During each of the seven visits, when the narrator shone a light upon the eye, he found it closed and thus relented from completing his insidious task. He obviously wanted the eye to be open before he proceeded. In his crazed state, the narrator clearly believed that the eye being open would be a signal for him to do his evil.

On the eighth night of the storyteller's surreptitious visit, he felt invigorated and proud of his intelligence and judgment. When he later shone the light on the sleeping man's eye, he found that it was open -- the sign that he had been waiting for for so long. He hesitated for a minute but then, disturbed by the sound of the old man's beating heart growing louder, went into a frenzy, jumped into the room, dragged the old man from his bed and pulled his heavy bed over him. The old man, clearly too frail and weak to push off the heavy object, suffocated and died. 

The number eight in the story also has symbolic significance: in Christian culture, it is seen as the number for regeneration and new beginnings. It is evident that the narrator believed being rid of the old man and, therefore, his terrible eye, would mean a new beginning for him -- a life without the horror of being exposed to the old man's evil eye.

It is something about the old man's "cloudy, pale blue eye" which incites madness in the narrator.  He says that when the old man looks at him, his "blood turns cold".  The narrator wants to see the "Evil Eye" once more before he kills the old man, but each night for the first week the old man is asleep when he sneaks into his room.  Finally, on the eighth night, the old man is awake.  The narrator shines a light on his eye and kills him.

Why does the narrator wait eight days to commit his crime in "The Tell-Tale Heart"?

The narrator in "The Tell-Tale Heart" is driven to murder, yet he isn't quite sure what drives him to do it:

He had never wronged me. He had never given me insult. For his gold I had no desire. I think it was his eye! yes, it was this!

Seemingly needing to convince himself that the man deserves to die, the narrator ultimately blames his plans for murder on an eye that is "pale blue" and covered with "a film."

He also finds that he cannot murder the old man while he sleeps. With the pale blue eye covered during the man's slumber, his motive for murder is extinguished. After all, he ruminates, the old man himself is not the issue—the "evil eye" is the source of conflict.

For seven nights, the narrator sneaks to the old man's bedroom door. He brags about how stealthily he approaches the room, taking an entire hour to place his entire head into a small opening so that he can observe the man's appearance. Using just a faint light from the lantern, the narrator ascertains whether the eye is open or closed, and for seven consecutive nights the "vulture eye" cannot be seen as the man soundly sleeps.

This changes on the eighth night. The narrator performs his ritual of observation and finds that the man has awakened, calling out, "Who's there?" as the narrator waits in the darkness, perfectly still and silent. When he finally shines the light of the lantern toward the old man, he finds that the eye is wide open, which makes the narrator furious as he looks upon it. This provides the motive he's been waiting for, and he kills the man in order to rid himself of the eye which vexes him.

The narrator's incredibly stealthy plans to murder a man because of the appearance of an eye point to his madness. It is also worth noting that while he feels he cannot kill the old man while he sleeps, he takes extreme precautions in order to avoid waking him for seven nights in a row. It would seem that if he needed the man's eye to be open in order to kill him, he would simply enter his room noisily. This points to an inner subconscious conflict within the narrator, demonstrating that his conscience struggles with his murderous plans.