The Quest Map if where you spend your Stamina and fight the computer.
There is a Main Quest line and several Portal Quest lines.
The Main Quest line currently includes 100 quests and Portals are 25 quests long. Portals are available for a limited time only so you will have new quests to play (and new cards to get!) every few weeks. All quests are numbered and their difficulty and reward increases linearly.
Every quest has two phases: Minion Fights and a Boss Fight
During Minion Fights, you defeat minor minions by tapping them. Minion waves vary from 2 to 4 minions, and after each wave, you have a chance to receive an entire card or gold. Each quest has 2-5 Minion waves, that need to be defeated to reach the boss fight. Fighting minions costs Stamina, and the necessary amount varies by the difficulty of the quest. Higher quests cost more Stamina, but also offer larger rewards.
In Boss Fights, you are pitted against one fixed card per quest and four random cards of a fixed quality. The boss fight does not cost Stamina or Mana. If you win the boss fight, you unlock the next quest and gain a reward: Gold, Gems, Potions, Tokens or a Card Piece. If you lose the boss fight, you will have to defeat his minions again!
You can always go back and replay any main quest you have beaten before.