Several pieces of equipment will mention challenges when looking at them in the inventory.
To complete the challenge, equip the item and complete its requirements, which could literally be anything such as stealth kill a captain or recruit an orc of x level or higher.
Once you have completed the challenge with the equipment active, an in game message will appear to inform you that it is complete.
Return to the inventory, highlight the item and press
to upgrade the gear by paying the small cost.
The achievement will then unlock.