- "You'll need milk of the poppy."
Milk of the poppy is a medical drink, used as both a painkiller and an anesthetic. Higher doses will induce unconsciousness, so patients can undergo surgery. It is commonly used throughout the Seven Kingdoms for those who have suffered severe injuries. Maesters make it from crushed poppy flowers and it has a white color, hence "milk of the poppy".
Milk of the poppy seems to have some sort of strong undesired side effects, probably some form of withdrawal, as some characters have preferred to go through strong pain instead of consuming it. Jaime Lannister refuses to take milk of the poppy when Qyburn was operating on the infected stump of his sword-hand, despite being warned that removing the rotting flesh would cause a massive amount of pain.
When King Robert Baratheon is mortally wounded after being gored by a boar while hunting, he asks to be given something to ease the pain before he dies. Eddard Stark then orders Grand Maester Pycelle to give him milk of the Poppy.
Eddard Stark is given milk of the poppy after a Lannister guardsman drives a spear through his leg. His daughter Sansa Stark later attempts to use this as an excuse for why he would openly denounce Joffrey Baratheon's legitimacy in court, saying that he had become feverish and the medicine he was taking had affected his behavior.
Talisa Maegyr comes to King Robb Stark's tent to warn him that her stockpiles of medical supplies for his army's wounded, including milk of the poppy, are running low. They decide to capture the castle known as the Crag to the west, so they can acquire their stockpiles of milk of the poppy.
After Maester Luwin is dying from a spear wound under the Godswood, Osha offers to bring him milk of the poppy, if he would tell her where he stores it. He declines the offer, revealing that his wounds are mortal, and asks instead that she give him the mercy of a quick death.
In the books
Milk of the poppy is essentially the Westeros version of morphine. Indeed, maesters make it from crushed poppy flowers, and morphine and other opiate-based drugs are made from poppy plants. The books' descriptions strongly suggest that the "poppy" is intended to be the real-world opium poppy (Papaver somniferum).
Milk of the poppy may result in some residual grogginess or withdrawal-like symptoms, which some characters would rather avoid by simply enduring the pain. Jaime rejects Qyburn's offer of milk of the poppy when he is operating on the stump of his sword-hand. Then again, in the books this chapter is narrated from Jaime's POV, which explicitly says that the main reason he refused to take milk of the poppy is actually because he feared that if he allowed himself to be rendered unconscious for the surgery, Qyburn might break his word and cut off the rest of his right arm anyway.