Bloodthirst is a Barbarian passive skill unlocked at Level 24, which causes the Barbarian to gain life based on the amount of damage he deals, in a traditional leech-skill.
This skill is set for a big change/nerf in Reaper of Souls. Datamined information tells of its future:
Bloodthirst: Each point of Fury spent heals you for 165 Life. Heal amount is increased by 1% of your Health Globe Healing Bonus.
This is part of RoS' across-the-board change to healing, which is almost entirely removing Life Steal from the game, whether it comes from skills or item bonuses.
One of the many powerful Barbarian passive skills, Bloodthirst provides a passive Life Steal bonus no other class can match. (Other classes can add Life Steal with specific skills, but never via a full time passive.) This skill makes things much easier for the Barbarian's survival and can open up more gear options, since Barbs can obtain Life Steal from this, or from their Mighty Belts, while all other classes can only get Life Steal from weapons.
There are no direct synergies with any active skills or other Barbarian passives. However all skills that boost damage boost the effectiveness of Life Steal, and the Barbarian has a number of those.
Passive skills have had a long and storied history. When Diablo III debuted at the WWI 2008, skill trees included passive skills, much like in Diablo II. In 2010, it was announced that passive skills had been renamed traits and separated out from active skills. At the July 2011 Press Event, Blizzard announced that passive skills were once again in the game, replacing the traits system.
Bloodthirst is unique in that it was an original passive skill and is once again a passive skill, but was never a trait. For the July 2011 Press Event, it had a listed level of 1 (presumably so players could try it out from the beginning), but for the beta, it was a level 10 passive.
The beta started in September 2011, and Bloodthirst was moved to level 10. When passives were reorganized for Beta Patch 14, Bloodthirst was moved up to level 24.
Players familiar with Diablo II (or WarCraft II or DotA) will recognize that this skill provides lifesteal. However, in Diablo II most skills didn't give life per hit, but there were health leeching prefixes on weapons, which could make any class gain health on hits.