Grand Maester Pycelle holding a glass bottle containing 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".
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.[1]
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.[2]
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 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.