Spellfire Tailoring is one of the Tailoring Specializations you are able to choose in Shattrath City.
Spellfire Specialization allows you to create two units of 
[Spellcloth] instead of one using the same amount of materials and grants access to Spellfire-specific tailoring recipies.
Becoming a Spellfire Tailor
To specialize in Spellfire tailoring, a character must complete 
[70] Becoming a Spellfire Tailor, given by Gidge Spellweaver <Spellfire Specialist> in Shattrath City. To obtain this quest you need to have 350 proficency in Tailoring, and be ready to travel to Netherstorm. You don't need to be level 70, but it's likely not soloable until you are.
To complete the quest, travel to Netherstorm and defeat the level 70 Nether-Wraith elemental that spawns when you use the quest item. You can summon the elemental anywhere in the map, but it is advisable to fight in a populated area, where someone might help you kill it. Area 52 guard NPCs will not engage the elemental. The elemental channels Arcane Missiles and it will cast a powerful Shadowbolt at low health. Although not elite, killing it solo is difficult for characters not yet 70, or those unprepared to fight it. Before engaging equip any arcane-resistance gear, if available. Mages should have up the Mana Shield, and save the Counterspell for the dangerous Shadowbolt, towards 20%-30% of the mob's hp.
The elemental will drop the quest item, travel back to Shattrath City to complete the quest.
You don't need any mats to complete the quest, and it won't start the cooldown on Spellcloth crafting. Immediately after completing the quest you may travel back to Netherstorm to craft your fist two pieces of Spellcloth.
Changing Specialization
Once you selected the Spellfire Specialization, you cannot change it. The only way to select a different specialization it to drop the Tailoring profession entirely, and start from scratch.
Crafting Spellcloth
Crafting Spellcloth is not exclusive to Spellfire Tailors, but the specialization will always produce 2 pieces of Spellcloth with the mats for a single one.
The materials are:
A character must be anywhere in the Netherstorm area to craft Spellcloth. The Nether-Wraith will spawn immediately after you end the crafting cast; you don't need to defeat it to keep the cloth, but since it always drops 2-3 Motes of Fire and 1-2 Motes of Mana, it is advisable to be prepared to defeat it.
The cooldown on the crafting is 96 hours (4 days), but it's not shared with the cooldowns on the other types of specialized cloth, namely 
[Primal Mooncloth] and 
[Shadowcloth].
Exclusive Patterns
Spellfire Tailoring gives exclusive access to the following patterns:
These patterns require the specialization to be obtained, and the items are Bind on Pickup, meaning only Spellfire Tailors could ever have the crafted items.
The Spellfire Set is arguably the best Fire/Arcane Mage/Destruction Warlock equipment attainable before the 25-man raid instances. It nicely complements the Spellstrike Set ([Spellstrike Hood]+[Spellstrike Pants]), because it uses different equipment slots.