There are 19 Pyromancies in the game. I will first list them in the order they appear in your inventory, then I'll explain how they can be obtained.
*** Spoiler - click to reveal ***Fireball
Fire Orb
Great Fireball
Great Chaos Fireball
Firestorm
Fire Tempest
Chaos Storm
Combustion
Great Combustion
Fire Surge
Fire Whip
Chaos Fire Whip
Poison Mist
Toxic Mist
Acid Surge
Flash Sweat
Iron Flesh
Power Within
Undead Rapport
Note that Black Flame from the Chasm of the Abyss is not required as it is from an area that was part of the original game's DLC .
11 can be purchased from NPCSIron Flesh: sold by Laurentius for 2000 Souls
Flash Sweat: sold by Laurentius for 2000 Souls
Combustion: sold by Quelana for 500 Souls
Fireball: sold by Quelana for 800 Souls
Great Combustion: sold by Quelana for 5000 Souls
Fire Orb: sold by Quelana for 8000 Souls
Undead Rapport: sold by Quelana for 10000 Souls
Fire Whip: sold by Quelana for 10000 Souls
Great Fireball: sold by Quelana for 20000 Souls
Firestorm: sold by Quelana for 30000 Souls
Note that some of these pyromancies are sold by Lauretius but all 10 can be bought from Quelana.
Toxic Mist: sold by Eingyi for 25000 Souls
Five are found or dropped by a specific enemyPower Within: found on a corpse in Blighttown near the Parasitic Wall Hugger enemy
Poison Mist: found on a corpse in Blighttown; sold by Eingyi for 10000 souls
Chaos Fire Whip: found in a chest in Lost Izalith near the Bed of Chaos fog wall
Acid Surge: found on a corpse in the Painted World of Ariamis near where Xanthous King Jeremiah invades
Fire Surge: 100% drop by an Engorged Zombie in the Painted World of Ariamis
Two are from covenant activity (joining or ranking up)Great Chaos Fireball: join the Daughter of Chaos covenant
Chaos Storm: offer 30 humanity to the Daughter of Chaos covenant
One is from an NPC questlineFire Tempest: given by Qualana after defeating the Bed of Chaos
NPC informationLaurentius of the Great Swamp relocates to Firelink Shrine after being rescued from the Depths. Tell him "no" when he asks if you find magics unsavory and he will give you a Pyromancy Flame. If you talk to him while possessing a pyromancy exclusive to Quelana, he will go mad and relocate to Blighttown where he will attack you on sight. Be sure to buy his two exclusive pyromancies before this happens.
Quelana of Izalith appears in Blighttown once you have upgraded your Pyromancy Flame to +10.
Eingyi can be found below the second Bell of Awakening behind an illusory wall that leads to the Daughter of Chaos bonfire and Quelaag's Sister. Tell him "yes" when he asks if you are a new servant and he will allow you to pass. In order to buy items from him, you must get infected by a hostile egg carrier enemy (allowing it to do a grab attack that stabs you in the head). After about five minutes, during which time your character will scratch their head, a giant egg will eventually bloom over your head. After talking to Eingyi like this, he will give you an egg vermifuge which can be used to cure the condition.
As always, refer to
http://darksouls.wikidot.com/ for specific information relating to items or locations.
This page gives specific information for each pyromancy.