Invocation can be used by all classes once an hour at a Campfire or Portable Altar to pray to the Gods (stand inside the radius and press "Ctrl+I"). It usually grants a 15 minute Bonus Damage and other possible rewards like Rough Astral Diamonds and Experience. Once a day, it will also give you a Celestial Coin and an Ardent Coin, which you can spend at the Vault of Piety for special items like boosts, dye packs or even mounts. Invocation is unlocked at level 11 and introduced through the quest Invoking the Gods, which can be started at Sergeant Knox or Aralynn the Pious. The quest is actually not required to use Invocation, but still recommended, because the HUD will only display the Invocation counter after the quest has been started, even if you have reached level 11.
The Invocation Counter is displayed in the upper right corner of the screen after you have started the quest Invoking the Gods. In the screenshot on the right, the counter shows that there are 39 minutes left before Invocation can be used again. It also shows that 2 Celestial Coins have been gathered so far. A left-click on the smaller circle on top of that number will open the Vault of Piety.