Chapter 26 of Harry Potter and the Half-Blood Prince: The Cave
Professor Dumbledore Apparates himself and Harry to the foot of a seaside cliff. This is where the young Tom Riddle led two younger orphans on a horrifying outing. As Dumbledore and Harry approach the cliff, Dumbledore illuminates his wand, revealing a fissure that he and Harry swim to.
Inside they find a sea cave, but it is only an antechamber; their actual destination lies further ahead. Dumbledore finds a slit in the cave's wall, but it does not open. Divining its secret, Dumbledore cuts his arm and sheds blood on the rock, explaining that Voldemort likely prefers to weaken those seeking his artifacts. An opening appears, and inside is an underground lake with a strange green glow in its center. As Dumbledore searches for something, he cautions Harry against touching the water. When Harry suggests Summoning the Horcrux Dumbldore urges him to try. Far out in the lake, something large jumps, intercepting Harry's spell. Dumbledore says that is likely what they will have to face in order to retrieve the Horcrux.
Dumbledore summons a submerged chain and reels in a tiny boat that has surfaced. Harry and Dumbledore climb in, and the boat propels itself towards a small island. Harry, peering into the water, sees dead bodies floating beneath the dark surface. Dumbledore assures him that as long as the bodies are merely floating, there is no danger, they will avoid light and warmth.
On the island is a basin filled with green glowing liquid. Presumably the Horcrux is immersed within. Dumbledore determines that he must drink the substance to uncover the locket and makes Harry promise to force him to finish every drop. He produces a goblet and begins drinking. The potion causes intense pain, and, in delirious agony, he begs Harry to kill him. When Dumbledore collapses, Harry persists in refilling the goblet and forcing Dumbledore to drink it all. Dumbledore finishes it, and, weak and thirsty, requests water. Harry conjures water, but it vanishes as soon as he gives it to Dumbledore. When Harry draws water from the lake, Inferi (the corpses floating beneath the surface, now animated) emerge and attempt to drag Harry into the lake. Harry tries fending them off, but there are too many. Dumbledore recovers enough to surrounded them with a conjured fire, driving the Inferi back into the water. After collecting the locket, he and Harry climb into the boat. The Inferi, dully, lose interest in Harry and Dumbledore, who return to the lake shore. Harry leads Dumbledore from the cave.
The only explanation we ever receive for Dumbledore asking Harry to accompany him on his mission to retrieve the Horcrux is that Harry "has earned that right," through his recovery of the memory from Professor Slughorn. It was actually critical that Harry go with Dumbledore, not only for his magical abilities, but also for his incomplete mastery of them. Dumbledore believes Voldemort charmed the boat to carry only one adult wizard at a time; it cannot be summoned to and from the island with magic while it is empty - a passenger must be in it. However, two people are needed to retrieve the locket from the basin. Harry is underage and small, so his presence in the boat goes undetected. If Dumbledore did not know this before he actually entered the cave, it is then fortuitous that he brought Harry with him.
It is a measure of Dumbledore's character that he deliberately undertakes the mission's hardest part. He refuses Harry's offer to provide the blood that opens the portal, and chooses to drink the unknown potion himself, rather than subjecting Harry to it.
- What was the green liquid that Dumbledore drank?
- Why would Dumbledore ask a still-unqualified student to accompany him on such a dangerous mission?
- Why did Dumbledore wish Harry to kill him?
Further Study 
- If, when Harry used Aguamenti to pour water into the goblet, it did not work, why didn't he simply attempt spraying a small quantity of water into Dumbledore's own mouth? He has seen Inferi in the lake, and, even if he doesn't know what exactly that means, he is frightened by it; why would he want to touch the water in the lake?
- We learned earlier in this book that Tom Riddle had taken two of his fellow orphans to a cave on a seaside trip, and they had never been the same after. We now learn that this is the same cave. Why would Riddle have brought the two orphans here?
- How could Tom Riddle have discovered the cave?
Greater Picture 
As mentioned, it takes two to recover the Horcrux, and one must be small and and unqualified wizard. It will soon be learned that Harry and Dumbledore are not the only ones who overcame this obstacle: Kreacher had been used by the Dark Lord for the same purpose, and had returned to his master afterwards, and Kreacher and his master had then also made the same trip.
The way this Horcrux was retrieved clearly indicates that Voldemort's power has limits, and demonstrates how his blindness to those limitations ultimately results in his defeat. This pattern is repeated throughout the series. In this case, Voldemort believes that his hiding place is impenetrable because he concealed his past history, forgetting that it was Dumbledore who retrieved him from the orphanage and that he probably knew about Tom's behavior at the orphanage. It is also apparent to us now that Dumbledore is familiar enough with Tom Riddle's thought patterns, that he is able to deduce how Tom conceals and protects his valuables. That the real Horcrux itself is gone is another piece of the same pattern; Voldemort, having protected his hiding place against wizards, discounts House-elves' magical abilities, and never considers Kreacher's knowledge about the cave to be an issue. Kreacher, of course, was unhindered by the anti-Apparition spells placed on the cave, and could return home after Voldemort left him for dead. This allowed Kreacher's master to learn about the Horcrux and to take steps against it.
It is worth noting Dumbledore's comment that Harry is more valuable than he is himself. Dumbledore now knows his time on Earth is very nearly done; it has been some eleven months since he was cursed by the Peverell ring, and Snape had given him no more than a year to live at that time. His time is even shorter than he can predict, as his death at Snape's hand is now only hours away. However, whether it is because of his own impending demise, or as a more general statement, we cannot be certain.
Compare Harry's feelings of "revulsion" and "self-hatred" as he forced Dumbledore to drink the potion, to Snape's expression as he performs the Avada Kedavra in the next chapter. Both Harry and Snape have been forced, by promises they have made to Dumbledore, to inflict harm on him. While Harry has accepted Dumbledore's promise out of love, it is uncertain what Snape feels for Dumbledore; we will discover later why Snape does Dumbledore's bidding, but not what his true feelings are. We will see in the next book that Snape continues to do Dumbledore's bidding even past his death, which suggests strongly that Snape did have some feelings for Dumbledore.