Demon blood is technically the blood of a demon's human vessel, effectively giving it supernatural properties. The blood from Azazel is responsible for creating people with special abilities in order to free Lucifer. Sam Winchester is one of these individuals. It once fueled Sam's psychic ability as well as his ability to exorcise demons. The more powerful the demon, the more blood must be consumed. Eventually, Sam consumed so much blood that his temperature and heart-rate went up drastically and his eyes turned black.
Demon blood saw its first appearance when Azazel was giving 6 month old children demon blood to create a future vessel for Lucifer (although this wasn't revealed until later). This blood soon created the Special Children and each developed their own abilities. It became known to Sam that he was fed demon blood in Season 2's All Hell Breaks Loose - Part 1 by Azazel.
Sam begins drinking demon blood when Ruby introduces him to the power it holds. Sam gained abilities such as exorcising demons with his mind and his abilities become increasingly powerful with every taste. Although humans were not meant to drink demon blood, only special children like Sam could and detox from it is quite painful. Sam is forced to detox after Dean discovers it in The Rapture and locks Sam in Bobby's Panic Room. However, Sam is so addicted to the blood that the detox is very painful, possibly even fatal to go through. When enough blood is drunk, it can permanently alter a person, although the effects are unknown. With enough blood, Sam can kill demons as powerful as Alastair and Lilith, but Castiel told Dean that the amount of blood Sam would have to consume to kill Lilith (an entire demon and presumably more blood from Ruby) would change him forever. Despite what Castiel said, Sam ends up being okay when God cures him of his demon blood addiction.
It is revealed that the host of Lucifer has to drink demon blood in order to hold Lucifer inside the body without bursting. Without his true vessel, Lucifer was forced to drink gallons of demon blood in order to keep his host's body, Nick, from burning up. When Sam prepared to let Lucifer in, he drinks about four gallons of demon blood taken from two demons he captured and killed. The amount of blood he drank made him so powerful he was able to kill two demons without moving a muscle. When facing Famine, his blood addiction was reawakened and he drank blood from demons sent after him. Despite Famine's overwhelming power, Sam managed to resist feeding on five more demons and exorcised them instead and later used this power to defeat Famine by exorcising the demons from him after he absorbed them. Sam had to detox from the demon blood again and survived.
The blood of the demon Crowley, the King of Hell was needed to make the Bone Of Righteous Mortal Washed In the Three Bloods of Fallen to kill a Leviathan. Dick Roman, the Leviathan leader, made a deal with Crowley for him to give the Winchesters the blood of the wrong demon so the weapon wouldn't work, but he gave them the right blood and let them know about the deal. Thanks to the blood, the weapon was completed and Dean killed Dick Roman with the help of Castiel.