The items that work for Currency are listed on that page, so won't be covered here.
Quest Items
There are Quest Items that are gained either by purchase from Shops or by completing Quests (oddly enough).
Custom Items
Custom Items covers both your House and your Character.
House
While you can purchase a handful of these from Events, or find them via Treasure Maps, these are primarily gained by earning the rarely found Custom Items from Map Zones.
Character
There are six categories here:
Only the Weapon gives benefits to your character, while the Pets will simply help you whack bushes a little faster. Everything else is purely cosmetic.