There are nineteen pyromancies in total. Equip a Pyromancy Flame in either hand and assign it to an attunement slot to use them. Once you have obtained them all one character, the achievement will unlock. The various merchants that sell them are below.

Eingyi
Laurentius of the Great Swamp
Quelaana of Izalith

Merchant Sold Pyromancies

Combustion - Sold by Laurentius/Quelaana.
Fire Orb - Sold by Laurentius/Quelaana.
Fire Tempest - Sold by Quelaana.
Fire Whip - Sold by Quelaana.
Fireball - Sold by Laurentius/Quelaana.
Firestorm - Sold by Quelaana.
Flash Sweat - Sold by Laurentius.
Great Combustion - Sold by Quelaana.
Great Fireball - Sold by Quelaana.
Iron Flesh - Sold by Laurentius.
Poison Mist - Sold by Eingyi after gaining his trust.
Toxic Mist - Sold by Eingyi after gaining his trust.
Undead Rapport - Sold by Quelaana.
Covenant Pyromancies

Great Chaos Fireball - Join the Chaos Servants covenant. See 'Covenant: Chaos Servant' for more details.
Chaos Storm - Reach rank 2 in the Chaos Servant Covenant. You need 30 humanity to do this, so you can farm the various rats in the Painted World, Depths, and Undead Parish. You can also farm the piasacas in the Duke's Archives.
Corpse/Chest Looted Pyromancies

Acid Surge - Treasure on a corpse in the Painted World of Ariamis. Turn right after the huge door after the bonfire, into the wide open area covered in impaled corpses and crows, where there are several torch-wielding Hollows and where Xanthous King Jeremiah invades your world. Turn right again, the Acid Surge is tucked away behind an outcrop.
Chaos Fire Whip - Found in a chest right before the Bed of Chaos battle. After the sister of chaos attacks you, continue straight rather than turning left and kill the Chaos Eater to find the chest.
Fire Surge - Dropped by a special engorged zombie in the Painted World of Ariamis. Go down the well into the dark hallway with Pinwheel skeletons, then when you get to the first junction, take a left. Follow that until you get to another junction and take a left again, then go up the stairs to find the engorged zombie.
Power Within - Treasure from a hanging corpse next to the parasitic wall hugger in Blighttown.