Demon blood is technically the blood of a demon's human vessel, which effectively gives 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. It was believed that the more powerful the demon, the more blood must be consumed in order to exorcise it. Eventually, Sam consumed so much blood that his temperature and heart-rate went up drastically and his eyes turned black. It was revealed in Lucifer Rising that Sam never needed to consume demon blood, and it was just a placebo effect. All Sam needed to kill Lilith and free Lucifer was the blood that Azazel gave him as a child. The blood that Ruby gave him was just to strengthen his vessel, so Lucifer could possess him after he set him free from his cage.
Demon blood saw its first appearance when Azazel was shown in flashback giving 6 month old children demon blood to create a future vessel for Lucifer (although this fact wasn't revealed until much 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 was forced to detox after Dean discovers Sam's habit in The Rapture and locks Sam in Bobby's Panic Room to "dry out." However, Sam was so addicted to the blood by this point that the detox was very painful, possibly even fatal to go through cold turkey. It was suggested by Castiel that when enough demon blood is consumed, it can permanently alter a person, although the effects are unknown. With enough blood, Sam could 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 ended up being okay despite the massive amount of blood he consumed, when God healed him of his dependence on demon blood so that he would not go through detox again.
It was 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 drank 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, he once again struggled with his cravings for demon blood as a result of Famine's powers, and ultimately succumbed to his cravings by drinking blood from demons who were 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.