"Fuse" is a game feature that allows you to get the blacksmith to "evolve" your equipment to a higher rarity. Only equipment that have reached their maximum levels can undergo fusion, and certain materials are required for this process. The materials required are listed in the item description window for each item, under the words "For Next Fuse".
Some items may have more than one material is listed, which means that the item can be evolved using either material, and the result will differ depending on which material is used.
If a "?" is shown below "For Next Fuse", this means that you have not yet encountered the material required to fuse the item. If you have picked up the material before but you do not have it in your inventory anymore, the material icon(s) will still appear under "For Next Fuse", but they will be darkened.
However, some equipment simply cannot be fused. This will be indicated in the item description window, where the word "None" will be shown under "For Next Fuse".
Any material of 1* or 2* rarity can be fused with an identical copy of itself to produce a material of one higher rarity. However, materials of 3* or higher cannot be refined in this manner. This means that 4* and 5* materials must be obtained directly, either from the market or by picking them up in dungeons.
Here are known evolution charts for the items in the game.