Moogle Shop
The Moogle Shop is a place that you synthesize items to make new weapons and armor, also it's a place were can be be buy items and cards.
Kingdom Hearts
In Kingdom Hearts it is only found on 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"; a special room generated by a map card 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, they synthesize items just like in Kingdom Hearts but here you also can buys items and weapons.