Once you reach level 60 and Tailoring 350, you may begin a quest to learn one of three specializations: Mooncloth Tailoring, Shadoweave Tailoring, or Spellfire Tailoring.
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Benefits of Specializations in WotLK
Any tailor with the required skill (415) can make 
Moonshroud, 
Spellweave, or 
Ebonweave. The benefit of each specialization is getting two pieces of that specialization's cloth with one set of materials.
- Mooncloth Tailor -> Moonshroud
- Spellfire Tailor -> Spellweave
- Shadoweave Tailor -> Ebonweave
This is the only benefit, there is no new BoP Tailoring set in WotLK Classic.
Specialization Quests
The three NPCs who start the quests are standing almost next to each other in Shattrat City, so it's pretty easy to spot them.
Mooncloth Tailoring
You can learn Mooncloth specialization by completing the 
Becoming a Mooncloth Tailor quest for Nasmara Moonsong.
To complete the quest, go to the Cenarion Refuge Moonwell (see the location below), open the 
Primal Mooncloth Supplies, and right-click on the 
Square of Imbued Netherweave to create a sample.
Zangarmarsh Moonwell
Shadoweave Tailoring
You can learn Shadoweave specialization by completing the Becoming a Shadoweave Tailor quest for Andrion Darkspinner.
To complete the quest, go to the Altar of Shadows in Shadowmoon Valley (see the location below) and use the 
Crystal of Deep Shadows quest item.
Altar of Shadows
Spellfire Tailoring
You can learn Spellfire specialization by completing the Becoming a Spellfire Tailor quest for Gidge Spellweaver
To complete the quest, go anywhere in Netherstorm and use the 
Nether-wraith Beacon quest item, kill the Furious Nether-wraith that spawns, then loot the 
Nether-wraith Essence.
Changing specialization
Changing Tailoring Specializations is very simple.
Go back to Shattrath City and talk to the NPC who taught your current specialization. It will cost 150g to unlearn the current specialization at level 70 and 20g to learn a new specialization. You don't have to do any additional questing.