There is plenty of equipment to be gotten in the world of Dragon's Dogma. You can equip weapons, which are restricted to the Vocation you are, as well as armor. All equipment can either be bought from merchants or found in chests and as drops from monsters.
Every piece of equipment can be upgraded a total of four times, although the fourth upgrade is special. This is represented by stars on the weapon, with the last, special upgrade shown as a flaming cross of some kind. Upgrading requires money and materials that can be gotten from enemies, chests and as rewards for quests.
The fourth upgrade is called Dragonforging, which means they are strengthened in wyrmfire. In order for this to happen, the item must be equipped by the Arisen, as unequipped items and those equipped by Pawns cannot be upgraded in this manner. All you have to do is kill one of the dragonkin, which are the Drake, Wyrm, Wyvern, The Dragon, and the Ur-Dragon. Note that the killing blow must be done by the Arisen and not a Pawn. Another note is that Pawns can equip Dragonforged equipment.
In the case of the Drake, Wyrm, Wyvern and offline Ur-Dragon, the piece of equipment must be upgraded to at least 1 star in order for it to have a chance of being Dragonforged (18% chance for three lesser dragons and 30% chance for offline Ur-Dragon; these are for 3-star equipment pieces). For The Dragon and the online Ur-Dragon, equipment doesn't have to be upgraded at all for it to be Dragonforged (100% chance).