The Red Thirst is, on the surface, quite similar to the Black Rage, in that it is a propensity for Blood Angels to fly into homicidal rages that compel them to get as close to melee as possible and generally wreck shit up. This similarity to the Black Rage has helped the Blood Angels obscure it, as it is one of their greatest shames.
Unlike the honorable Black Rage, the Red Thirst is comparatively short-lived, making it possible for a Blood Angel to ride it out and not become a mindless killing machine (whereas the Black Rage is an incurable mental trauma that leads to inevitable death). Though sometimes the opposite is true. It stems from incredibly intense and violent urges towards haemovoria (blood-drinking) and cannibalism, which makes Blood Angels and their successors who succumb prone to devouring their victims when there are no more targets to slay. More so than the Black Rage, the Red Thirst blinds a Blood Angel to the nature of those around them; any living creature, whether xenos, traitor, heretic or loyalist, becomes a source of blood.
As a result, Blood Angels strive to hide this trait from Imperial eyes, lest they be purged as deviants. The Flesh Tearers chapter is particularly strongly grasped by the Red Thirst, resulting in their gods-awful reputation due to their propensity for slaughtering civilians and allies as readily as their enemies, whilst the Blood Drinkers have embraced the Red Thirst to a degree that disgusts their cousins.
Exactly what causes the Red Thirst is unknown, but it is believed to be aggravated by the Blood Angel practice of sanguinary transfusion - using their Apothecaries to host the living blood of Sanguinus and to siphon this from them into new recruits as part of the transformation.
Originally, it was stated that this is just one of the ways that the Blood Angels have degenerated since the Great Crusade, but the Horus Heresy novels have retconned that the Red Thirst was beginning to surface even then. In addition to laying solid hints that the Missing Chapters were exterminated for mutation, it also jossed a long-held fan theory that, were Sanguinus to be restored, he would be able to purge the Red Thirst from his legion.
Of particular note and interest to the Red Thirst is the Omophagea organ, which is one of the nineteen organs that make up a Space Marine's Gene-seed, gives the body the ability to learn and experience things of the things that they've eaten.
Another thing of note is that in its most advanced stages, the Red Thirst seems to have mutating effects. In the comic "Bloodquest", the Marine known as Cloten begins developing horn-like structures on his head due to the Red Thirst, and has a vision of himself degenerated into a hulking savage with a bat-like face. This is corroborated in "Devastation of Baal", where the marines lost to the Red Thirst are released from the Tower of Amareo and are noted to have devolved into monsters twice the size of a man that walk on their knuckles like a gorilla. It is also stated in "Devastation of Baal" that the Knights of Blood, who largely succumbed to the Red Thirst before dying fighting off Tyranids and Daemons to save the Flesh Tearers, had their visage similar to Ka'bandha with bulging muscles under their skin, red skin, fangs instead of teeth while stretching their lips, and amber eyes.
On the tabletop, the Red Thirst has been revised with basically every edition. In 8th, it's amazing: Blood Angels infantry, bikers, and Dreadnoughts receive +1 to all wound rolls in the fight phase any turn they charge, get charged, or perform a Heroic Intervention, for maximum rip and tear. This has even the most basic Blood Angels infantry wounding everything in the game on a 5+, and hurting most infantry on 2+ or 3+ and most vehicles on a 4+.