The original Carnage symbiote was the first offspring of the Venom symbiote, spawned when it broke Eddie Brock out of prison. The newborn symbiote bonded to serial killer Cletus Kasady, merging with his blood to become nigh-inseparable from him, and Carnage was born.
The Carnage symbiote has been separated from Cletus on numerous occasions, leading to it temporarily bonding to other individuals to give rise to Spider-Carnage, Carnage Cosmic, Superior Carnage, the Red Goblin, and the Goblin Childe but each time Cletus either managed to regenerate it from remnants in his bloodstream or reunite with it. This has also resulted in the existence of several duplicates of the Carnage symbiote, some of which have been taken into the custody of the US military or corporations like Alchemax.
Cletus has also been bonded to other symbiotes -- one which he found in the Negative Zone, and which was seemingly assimilated by the remnants of the original Carnage symbiote in his body to rejuvenate it; and one which was forcibly bonded to him by the Poisons and then consumed to give rise to Poison Carnage, ultimately perishing in orbit after the death of the Poison Queen.
Following this, the remnants of the original Carnage symbiote within Cletus' body resurfaced in an attempt to shield him from atmospheric re-entry, with both of them perishing in the process; and Cletus' corpse was bonded to an offshoot of the deific Grendel symbiote dragon; imbuing it with the memories and personality of the original Carnage symbiote to create Dark Carnage.