Rare items (items displayed in inventory with a blue, purple or gold background) may be reforged to increase their power score (PS).  Reforging requires coins, shown in the bottom left, and one or more specific smith's hammers, shown in the left hand panel.  Hammers are found as loot drops in combat, or may be purchased from the store.

Note that reforging does not gurantee an upgrade will be successful. The right-hand panel displays the maximum that your reforge attempt may produce.  There is an element of random chance in the process.  You may discard an update and try again if you feel a need to. When you reach a level with which you  are satisfied, tap "save" to accept the upgrade.

If the maximum upgrade possible  is the same as the current PS, the item is not capable of being upgraded further.