Upgrade items to improve their stats or unlock their unique features (you can check if any unique features exist in the item description, below the item stats).
You can upgrade any equippable items you own, except for grenades and consumables.

You must first complete the tutorial (it starts after the "The Drop-Off" story mission). Once completed, you can upgrade at any time (if you're not in a mission) by tapping the Upgrade icon in the Shop.

You will need to acquire MATERIALS to upgrade (there are 4 types of materials and 3 levels of quality per type). You can check the requirements at the bottom of the upgrading menu.

If you don't have enough materials, you can open Materials Packs in the same menu or in the Crates section of the Shop. You can also trade in the Materials you don't need right now for other ones using Exchange Packs, which you can find in the Shop.