A Completion Upgrade is an Item which allows you to Upgrade your buildings to a Lvl. 10.
You can only use it on a building that is level 9. there are no shortcuts.
Without these, your buildings will be stuck on Lvl. 9.
Also, Some things are not available to you until you use a upgrage. For instance, you need a Lv. 10 Barracks with your outpost's to start the training of Alchemest and Reapers.