Demonic Organ

Demonic Organs are special materials used to craft the Hellfire Ring. There are three of them in Diablo 3 and four different Organs in Reaper of Souls. Each of the Organs is dropped only by certain of the Ubers, the special boss monsters reached via the portals in the Hermit's Shack.

Demonic Organs and all other components of the Infernal Machine are Torment only in D3v2 or Reaper of Souls, and do not exist or function on lower difficulty levels.

See the Infernal Machine article for full details on how to obtain keys, create Infernal Machines, open the portals to the Ubers, obtain Demonic Organs, and use those to craft a Hellfire Ring. The process is roughly the same in Diablo 3 and Reaper of Souls, though RoS adds a 4th Uber and a 4th Demonic Organ.

Diablo 3 Demonic Organs

Higher difficulty settings increases their drop rate, from 25% on Torment 1 up to 50% on Torment 6.

The Writhing Spine drops from King Leoric or Maghda in the Realm of Discord.
The Devil's Fang drops from Ghom or Rakanoth in the Realm of Chaos.
The Vengeful Eye drops from Siege Breaker or Zoltun Kulle in the Realm of Turmoil.

All three organs are required to craft a Hellfire Ring in Diablo 3.

Reaper of Souls Demonic Organs

The Writhing Spine drops from King Leoric or Maghda in the Realm of Discord.
The Devil's Fang drops from Ghom or Rakanoth in the Realm of Chaos.
The Vengeful Eye drops from Siege Breaker or Zoltun Kulle in the Realm of Turmoil.

Demonic Organ Drop Rates

The same odds apply to Keys from the Keywardens. These figures are for D3 or RoS, after D3v2 completely changed around the difficulty system.