In total there are 4 different game modes, these are League Match, Room Match, Training, and Quests.
Training
These are practice matches that let you earn Gems and learn advanced techniques.
Quests
This is the story mode of the game that allows you to earn Exp., Coins, and more by battling A.I. opponents. Doing quests also gives you the opportunity to earn extra Gems by completing challenges in each duel.
League Match
These are matches that match players of similar ranking and will affect your League rating.
Room Match
These are matches you can create or join that don't affect your League rating.