Cards become stronger when leveled up. Copies of the same card and gold are required to level up a card.
Cards that can be leveled up will have a full card gauge on the Deck screen. To level up a card, select the card and tap the "Lv. Up" button that is at the bottom right on the details screen. When a card level up, some of its stats increase. The same stats will increase at the same percentage with each level.
Leveling up a card also provides Player EXP., required to increase your Player Level.