Some items cannot be repaired. This is stated within the Item Description of your item. On the other hand, some items can only be repaired at certain shops or NPCs. Make sure you’re talking to the right NPC!
To quickly find out where to visit, or what repair type your item is, view the Item Attributes of your item by mousing over the item in your inventory.
The devCAT Brooch is an accessory item, which can be repaired it the Bangor Pub. Talk to the NPC Jennifer to bring up the repair menu. This is listed by the attribute, "Bangor Pub Repair".
As another example,
Clothing items will list their repair location as "Clothing Shop Repair".
Some items are special, and must be repaired at a specific NPC. Look out for items that list the attribute "Fleta Repair", as you'll have to find the NPC Fleta to repair your item. Fleta in located in Sen Mag Plains, and will spawn at different times per in-game day.
A placed Altar of Restoration can also repair all repairable items in your inventory. Altars of Restoration can be purchased from General Store NPCs, such as Walter in Dunbarton or Effie in Qilla Base Camp.
Once purchased, 'Use' the item and choose an appropriate location to place the Altar of Restoration into the world. Not all locations are suitable for the Altar.
Interacting with the Altar of Restoration will bring up the repair interface, from which you can repair your items! Note that the Altar will allow you to repair any repairable item, regardless of the NPC that the item would normally be repairable at!