Many items can be refined. Refining raises a single stat on your item. The maximum level you can refine to is level 11.
If you wish to refine an item, visit [NPC] Blacksmith Marx. Blacksmith Marx can be found in various places, such as Paradise Shop. Talk to Blacksmith Marx and select Refine Item. Put your item of choice in the Item Slot and you will see what material is needed for refining, the increase of one stat, the percentage of succeeding and the refinement fee.
If your items breaks through refining, and is not a MyShop item, it cannot be equipped anymore. In this case, you will need Repair Powder from MyShop, to repair your item. To find out how many Repair Powders you’ll need, talk to Blacksmith Marx, put your broken item in the Item Slot, and he will tell you.