To upgrade a card, tap on it in the BUILD MENU to open the card details, then tap on the UPGRADE button.
Each upgrade increases the card's STATS and therefore its PERFORMANCE in the park.
Each upgrade requires a certain AMOUNT of cards. Be aware that the next upgrade for a given item will require more cards.
Note that when you upgrade a card, all the corresponding items placed in your park are upgraded.
Tap on the UPGRADE button to proceed. 
Upgrading a card also allows you to place more copies of the same item in your park.