The Armory is where you buy Common (White) equipment, upgrade equipment you own, access the Emporium, and sending out Scouts for Mounts and Cloaks (scout feature unlocked at level 30).
The success rate for upgrading equipment changes every 30 minutes. It rises towards 100%, then lowers towards 50% and repeats. It can change by small percentages like 5-10 or by high percentages like 20-30.
The Armory's level is the limit you can upgrade equipment. Meaning that awesome first green weapon you get can only be made strong by upgrading your City Hall, followed by your Armory, and finally upgrading the item.