The goal is to reduce the enemy hero's health down from 30 to 0. You play minions or spells which damage or control the enemy's minions or hero. These cards cost mana to play. You gain one mana crystal per turn until you are at 10 mana crystals, and each turn your mana is refreshed to full. There are 9 classes you may play as in Hearthstone and I believe they are the vanilla World of Warcraft classes - Mage, Priest, Warlock, Hunter, Druid, Warrior, Paladin, Rogue, Shaman. Each hero has their own specific hero power that makes them unique.
This video is somewhat dated but it gives a good overview of the game:
The other main game mode 'Arena,' costs gold or real money (currently $2) to enter. In this game mode you can choose between 3 champions randomly presented to you. Then the system shows 3 cards to you and you must pick one to add to your deck. Then it shows 3 new cards for you to choose from, and repeats until you reach the 30 card deck. Then, you play in the Arena against opponents of your skill level using that fixed deck you created until you lose 3 games. Once you lose 3 games that deck is gone forever and if you want to play in the Arena again you must pay to re-enter. The benefit is though that you get nice awards from the Arena that scale depending on how many wins you get with that deck.
There is also a practice mode to play against AI opponents but you get minimal rewards for doing that and it isn't really the point of the game. It is a nice introduction to the game though.
At first I thought Arena was my favorite mode because I liked the element of it that chose cards for me, but now I'm really getting into building my own decks and fine tuning them over many games to do what I need, so I've been doing the ranked Play mode more often. The Arena is a one shot go, so if you build a bad deck you just need to suck it up and do what you can. The awards for the Arena are great though so it's still tempting to play.
So far this game is pretty fun and I quite like it. I will write two more sections about Hearthstone. The next installment will be basic strategy, and the final installment will be my criticisms and suggestions.