In Azur Lane you can upgrade equipment to make it stronger, which is almost always to your benefit.
Upgrading gear can be done from two different menus.
This first and most helpful way is though the dockyard menu. The second is though the equipment menu on the main screen. The menu though the dockyard menu can be accessed by Clicking on the blue button labeled "dockyard" on the bottom left. then clicking on the ship card that has the equipment you want to upgrade. You will now be on the Ship Equipment screen.
From here, on the right highlighted in red is your ship equipment. Clicking on a piece of equipment will bring up this menu:
The gold button on the right is the upgrade button, the button in the middle is for switching equipment, and the button on the left removes the current item in the slot you clicked. On the top right of this screen is your equipment tier(in the above image it's "T1"). You should also take note of the background color on the icon, this represents its rarity. Upon clicking the gold button, you should be greeted with a menu similar to this one:
The area highlighted in blue are the materials you need to upgrade the current item you have selected. Most upgrades require tokens, tokens are weapon and rarity specific and cannot be used on anything but the weapon and rarity they are assigned to. Some low-level upgrades on rare items will allow the use of low-level Tokens to the tokens rarity level. eg. A common token can be used to upgrade a rare item to level 3 (the common limit), then the equipment will need the next rank up of token to be upgraded further. Tokens can be acquired via missions under the "task" tab and sorties under the button labeled "weigh anchor" map. Tokens are consumed upon use and need to be farmed.
Highlighted in red is the stats prior to and after the upgrade. In the image its the damage and number of bullets, and underneath that the the shot cooldown. The green text on the right-hand side is the change in stats (what you will gain by upgrading a piece of equipment).
The amount of times an item can be upgraded is dependent on its rarity. Tier one common Equipment can only be upgraded 3 times, however higher rarity equipment can be upgraded beyond this.
The gold button to the bottom right is the upgrade button, and above it is the amount of money the upgrade will cost after the requirements are met.
Upon successfully upgrading a piece of equipment the icon will change, assuming it is the first time you have upgraded it, it will have a yellow "+1" in the top left corner. This number will get bigger per upgrade.