Trading cards are items that trigger events in the game. Those events can make cosmetic changes, or they can give you small advantages thoughout the game.
With some of the trading cards you can change your match settings at half time to influence the course of a match, you can improve your players' talents, or add extra training slots.
In the card shop you can buy standard booster packs (7 random cards per pack) at any time, or purchase specialized booster packs that are not always offered. You can also purchase cards in auctions on the bazaar or directly from your card deck. And you can win cards on the goal wall.
To take a look at your cards you can open your card deck.
Check here for a list of all available cards.