Paid Gems are Gems acquired through the in-app purchases in the game via Gems Shop.
Starting from Jan. 10, 2018 PST, an update was implemented in order to improve the management of our premium currency. After the update, the system shows Bonus Gems and Paid Gems under player's Gem Summary. To see the announcement details, kindly refer to this LINK.
You can access your Gem Balance Summary via the Summary button from Shop > Purchase Gems screen, or by tapping and holding on the Gem Balance interface located at the top right corner of the game screen.
To know more about Bonus Gems, kindly click this LINK.