The Moogle House can be used for many things, like:
Regain your hp and mp
Store items you don't need right away or don't want to accedently use and will if you have it
Change your job
finally, you can buy items for the traders in the Mog House
Where do I find the Mog House!?
The Mog houses are found in most major cities and towns
The mog house will be found in the residential area of the town