Each of the nine classes has different decks that give you different abilities. You'll notice that many classes don't yet have the 20 earnable cards -- you can play real players or practice games to fill out those decks.
Once you start the game, you have the option to replace cards in your opening hand. The level (and crystal requirement) for the cards are in the upper left of each card. You might replace high cards here so you can play more in the early game, but some of those high cards might be really important later.
As you play, you can lay down minion cards that help in both attacking your opponent and defending your hero.
Once you've whittled away at your opponent's hit points, you'll hopefully have the opportunity for a deathblow that makes his hero explode and makes you win the game.
As you play, you'll earn cards by winning and you'll also be able to craft cards to add to your deck. Custom decks I have already made are listed on the right.
Go to the Quest Log to see your stats and check out the ongoing quests listed at the bottom. Each completed quest earns you gold, which you can use to buy more booster packs of cards.