Socket Gem

Gem Socket Academy is a place where you can socket gems on your gears. Before this, the only thing you should make sure is that there is a socket on the gear.
There are 4 types of gems: Potency Gem, EXP Gem, Attack Gem and Defense Gem. Each type of gem can only be socketed in some specific gears. 

Put in the socketed gear and the corresponding gem, then click Socket.

After that, check your item's description, you can find more details.

Remove Gem

Put in the gear that has been socketed with a gem and click Remove.

Keep in mind that after being removed, the gem will disappear.