Moogle Shop
The Moogle Shop is a place that you synthesize items to make new weapons and armor, as well as a place where items and cards can be bought.
Kingdom Hearts
In Kingdom Hearts it is only found in Traverse Town. When Sora uses the green Trinity Mark in the Accessory Shop in the First District, the Synthesis Workshop which is run by some of the Moogles is opened and Sora can then go there to allow the Moogles there to use some of his items to make new ones.
Kingdom Hearts: Chain of Memories
In Kingdom Hearts: Chain of Memories it appears in "Moogle Rooms"; special rooms generated by map cards of the same name. Inside, Sora is able to cash in his unwanted cards for "Moogle Points" and is able to spend accumulated Moogle Points on purchasing new card packs.
Kingdom Hearts II
In Kingdom Hearts II, they appear in every world, but most of them are run by holograms of Moogles. Just as in the first game, items can be synthesized in Moogle Shops, but the player can also buy items and weapons.