When Snape climbs the steps up to the Astronomy Tower in Half-Blood Prince (The Lightning-Struck Tower chapter):
But somebody else had spoken Snape's name, quite softly. "Severus..." The sound frightened Harry beyond anything he had experienced all evening, Dumbledore was pleading.
And then
“Severus . . . please . . .”
Was this to keep up the illusion, or was there unfinished business, the plan was not ready that night? Did Dumbledore not want Harry to see?
Snape was Dumbledore's double agent from the beginning and the plan was for Snape to kill Dumbledore (who was dying anyway) to prove his loyalty to Voldemort, which would result in him getting closer. Snape was against this plan but had reluctantly agreed. When the time came for Snape to actually kill Dumbledore he hesitated, which is why Dumbledore pleaded with him. Not to spare his life, but to do his part in the plan and kill Dumbledore.
The other answers here aren't wrong. But they don't contain the (whole) correct answer. Namely:
This is what causes Dumbledore to plead with Snape. He knew that he was going to die one way or another. He had embraced death and wasn't scared of it. But he was worried that he might die at the hands of another Death Eater (like Greyback) who would cause his death to be drawn-out and traumatic.
"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 favour to 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."
His tone was light but his blue eyes pierced Snape as they had frequently pierced Harry, as though the soul they discussed was visible to him. At last Snape gave another curt nod.
(Deathly Hallows, Chapter 33, The Prince's Tale).
Note that Dumbledore wants to avoid pain and humiliation and is specifically worried about Greyback and Bellatrix. When he's disarmed on the Astronomy Tower Dumbledore knows that his time must be up. Either Snape will kill him or one of the other Death Eaters will. Bellatrix isn't there to do it but, as he had feared, Greyback is.
"Is that you, Fenrir?" asked Dumbledore.
"That's right," rasped the other. "Pleased to see me, Dumbledore?"
"No, I cannot say that I am..."
Fenrir Greyback grinned, showing pointed teeth. Blood trickled down his chin and he licked his lips slowly, obscenely.
"But you know how much I like kids, Dumbledore."
"Am I now to take it that you are attacking even without the full moon now? This is most unusual...you have developed a taste for human flesh that cannot be satisfied once a month?"
"That's right," said Greyback. "Shocks you, that, doesn't it, Dumbledore? Frightens you?"
"Well, I cannot pretend it does not disgust me a little," said Dumbledore.
[...]
And he raised a yellow fingernail and picked at his front teeth, leering at Dumbledore.
"I could do you for afters, Dumbledore..."
(Half-Blood Prince, Chapter 27, The Lightning-Struck Tower).
Dumbledore was brave enough to stand up to Greyback. He wasn't going to betray his fear openly in front of the Death Eaters. But that doesn't mean he wasn't scared. He is unarmed and defenceless. Greyback is demonstrating his very real desire to rip him to pieces. That's not a pretty death, as Dumbledore himself says in the earlier quote. Although he also cared about saving Draco's soul, as Dave Johnson says, he is really speaking out of self-interest here. Snape is his one way out of a grisly, painful death. Yet Snape was unwilling to kill Dumbledore - perhaps out of love/respect for Dumbledore and perhaps because killing someone is simply a big deal (Snape has to summon the will to go through with Dumbledore's request). So Dumbledore pleads with him to keep his word and finish him off - knowing the fate that awaits him if Snape should fail.
It was revealed that the plan the entire time was for Snape to kill Dumbledore, because Dumbledore was foolish when handling one of the Horcruxes, and thus dying anyway. Having Snape be the one to kill Dumbledore ensured that Voldemort would trust Snape that much more.
Also, more specific to the question, Dumbledore was pleading because above all else, Dumbledore did not want Draco to be the one to kill him. Dumbledore very much believed that Draco could be saved, and knew that killing someone fractures your soul. In fact, that is specifically what must be done to create a Horcrux:
By an act of evil -- the supreme act of evil. By committing murder. Killing rips the soul apart. The wizard intent upon creating a Horcrux would use the damage to his advantage: he would encase the torn portion. ...
I think that Dumbledore said "Snape, Please" since he wanted him to carry out the effect and make an effect that made even Harry think Snape was evil. What do you think?
Dumbledore was actually telling Severus to go along with the plan, that is, killing him and thus gaining Voldemort's trust.
Saying "Severus... please", was Dumbledore pleading to him, like: "Snape, you gotta do it."
