Currently we there are two different challenge modes: Random & Endurance.
Challenge mode is a basic fun only mode. You are pitted against a random player. A chance to practice your deck without consequences, but also without any reward.
Endurance mode is not for the feint hearted. The enemy boss has 999 HP, kills troops that get too close, and continually gets his deck bolstered. Fortunately the goal is not to win, simply survive as long as you can before you are defeated. Your kill score is displayed in the bottom left corner and Blue Gems are awarded when you are defeated. The more kills you make the more gems you get.