Draco just disarmed Dumbledore. He ultimately could not bring himself to kill him, even after the arrival and encouragement of several Death Eaters, who were smuggled into the castle by Draco.
To save Draco's soul, Dumbledore ordered Snape to give him a quick and painless death when the time came.
“If you don’t mind dying,” said Snape roughly, “why not let Draco do it?” “That boy’s soul is not yet so damaged,” said Dumbledore. “I would not have it ripped apart on my account.” “And my soul, Dumbledore? Mine?” “You alone know whether it will harm your soul to help an old man avoid pain and humiliation,” said Dumbledore. “I ask this one great favor of you, Severus, because death is coming for me as surely as the Chudley Cannons will finish bottom of this year’s league. I confess I should prefer a quick, painless exit to the protracted and messy affair it will be if, for instance, Greyback is involved — I hear Voldemort has recruited him? Or dear Bellatrix, who likes to play with her food before she eats it.”
The act of committing murder causes the murderer's soul to become damaged. It seems that the murder may be committed indirectly as long as the murderer has the intent, seeing as how Tom Riddle used the Serpent of Slytherin to kill Myrtle Warren to create his first Horcrux rather than kill her himself.
The Killing Curse requires a genuine willingness and desire to commit murder. One of the main reasons why Voldemort demonstrated such an affinity for the curse was due to how exceptionally powerful he was magic wise along with his complete and utter lack of remorse or value for the lives of fellow humans, Muggle or otherwise. Draco, despite possessing many undesirable personality traits, found himself ultimately unable to kill Dumbledore because he did not hate him enough to actually do the deed. Voldemort, on the other hand, had no such restraint and murdered scores of people without remorse in his pursuit of power and immortality.
It should be noted, however, that killing in and of itself seems not to have the same consequences to a soul as committing deliberate murder, as Dumbledore seemed to imply to Snape when requesting that Snape kill him, that the act would not harm Snape's soul due to the circumstances of the killing.