Dumbledore asked Snape to kill Dumbledore. Over the summer between the 5th and 6th books, Dumbledore went looking for the ring Voldemort had stolen from his grandfather Marvolo Gaunt (well, actually, he stole it from his uncle Morfin Gaunt, who inherited it after Marvolo died), and which had originally belonged to Salazar Slytherin. Voldemort had made the ring into a Horcrux and put a curse on it.

What Voldemort didn't know was that the black stone in the ring was actually one of the Deathly Hallows -- the Resurrection Stone. Dumbledore had never gotten over the death of his baby sister Ariana, who was killed by a spell during the fight Dumbledore's brother Aberforth had with Dumbledore's friend, Gellert Grindelwald. Even though he knew the ring was probably cursed, he just couldn't resist touching the stone, because he wanted to see his sister again so badly.

Dumbledore immediately found Snape, and Snape was able to contain the curse in Dumbledore's hand. If Snape hadn't done this in time, the curse would have spread to the rest of Dumbledore's body and killed him fairly quickly. But even though Snape was able to contain the curse, Dumbledore was still doomed: he had only about a year to live.

So, since Voldemort was already planning to make Draco Malfoy kill Dumbledore, Dumbledore decided to ask Snape to kill him instead. Dumbledore knew that Voldemort didn't really expect Draco to succeed at killing him; Draco's parents had fallen out of favor with Voldemort, and Voldemort just wanted to heap more misery onto the family, by making them watch their son fail at the task and be killed for it. And even if Draco was able to kill him, Dumbledore didn't want Draco's soul to be damaged by murder. Also, if Snape killed Dumbledore, it would seemingly prove his loyalty to Voldemort, and ensure his continuing worth as an adept and loyal Death Eater.