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What is Hearthstone?
Hearthstone is a free-to-play online Collectible Card Game (F2POCCG, which is a real genre) from Blizzard, based on the Warcraft universe. Choose a hero corresponding to one of the nine classes in Vanilla WoW and create a deck using their signature mechanics to fight other players in Casual or Ranked matchmaking. Earn Gold for every three wins and use the Gold to buy booster packs in order to expand your card pool, or use the Gold to purchase a ticket to the Arena where you construct your own deck on the fly using a random pool of all cards in the game.
Download the client (Windows, Mac) on the main site. Note, you will also need a Blizzard account to log in to their launcher.
How to play Hearthstone?
I'm not going to go into that here, just play the tutorial to learn the game.
How to really play Hearthstone?
Read articles, apply critical thinking, watch good streamers like Trump.
Is there an in-game/player-driven economy?
No. There is no card trading or buying cards from other players and there probably won't be anytime in the foreseeable future. In its place is a crafting system. You can disenchant cards you do not want for arcane dust. You can then use that arcane dust to craft cards you do want.
Is this game Pay2Win?'
Of a sort. The thing you should know about Hearthstone is that if you're not going to pay money, plan on playing a lot (read: grinding) or becoming really good at Arena. The longer a player has played, the bigger their card pool will be. This means that an older, frequent player will always have better cards than a newer player. This is just an unavoidable part of collectible card games.
A paying player can save a lot of time by buying boosters, but it's not cheap and they're not guaranteed the best of the best cards, which is, in the end, what separates good decks from the best decks. Buying some boosters right off, however, will help you build a foundation to reliably win games in order to get Gold.
If you really hate not having good cards, then stick to Arena, which is a "level" playing field for a certain definition of "level".
What's the balance like?
Eh. If you're used to the Blizzard cycle of buffs and nerfs and FOTM, you'll feel right at home here. Because of the digital format, the dev team can play a little fast and loose with card balance compared to, say, Wizards of the Coast. But this also means they make more mistakes, and overall they're still trying to find that card balance sweet spot. Once they do find their footing, however, they will dial back the number of changes and generally take a hand off approach.
If I like MTG/YGO/blah blah blah, will I like this?
Hearthstone is simplistic compared to games like MTG and YGO. The reason for this is because of Hearthstone's asynchronous turns, which means that you can't make any moves on your opponents turns. Hearthstone does have it's own particular brand of complexity, but it's really designed from the ground up to be a "pick up and play game". If you're looking for a hardcore card game experience like MTG, you won't find it here, but if you're looking for a streamlined card game experience you can jump in and out of whenever you want, then Hearthstone is right for you.
Anyway it's free, so you don't lose much by giving it a shot.
Le Reddit Guide - Probably the best place to start if you're totally new.
Liquid Hearth's Beginner Guides - Some of the info is outdated but the general ideas remain the same.
Three Competing Resources Parts 1, 2, 3 - A series of articles on "resources" in the context of a card game, which goes well beyond just mana.
Official Forums - A veritable nest of QQ and cries for nerf, just like your WoW days!
Team Liquid's Hearthstone Portal - Better place to discuss and learn about Hearthstone, frequently featuring competitive analysis and deck tech articles.
IHearthU - Another good site for high level Hearthstone discourse.
Hearthpwn and Managrind - Part of the Curse family of gaming sites, not only does it have an online database of cards and deck building tool, it is also home to ManaGrind, one of the premiere Hearthstone tournaments!
Hearthstone Wiki - Obligatory game wiki.
Trump: YouTube, Twitch
Value Town and Deck Wars (Chanman): YouTube, Twitch
Funny and Lucky Moments: YouTube
2pPressStart; YouTube
Hearthstone Calendar
Team Liquid Calendar