The Forge is a building where you can collect various equipment produced daily; it also gives you an option to forge your own equipment based on what you have in your inventory. Forging equipment is a Daily Mission requirement that shouldn't be skipped!
The Forge is available at Castle level 2, and has a medium-level of wait time when you upgrade it. Equipment in general has more impact on a Heroes' strength at the start of the game rather than later, so it's worthwhile to try and upgrade this building as soon as possible in the early stages.