I loved Magic the Gathering: Online, and while it definitely had issues, Hearthstone corrects many of those issues and invigorates the genre. I believe that Hearthstone is to Magic Online what World of Warcraft 1.0 was to Everquest -- a good spin on an old genre.
Hearthstone uses the flavor of the Warcraft universe for the theme, and it's not hard to envision other variants using other Blizzard properties.
My main gripe with Hearthstone is that it's very turn based; unlike other Blizzard games where you're constantly in the action, Hearthstone is turn-based, and doesn't have responses the way MtG does with Instants -- the only Instant like action is in the form of secrets, which only three of the eight classes have and auto-execute when conditions are met; while this speeds up Hearthstone considerably (no more waiting for the other player to see if the move is ok, like in MtG), it also makes it so that there's no real reason for the player to pay attention when it's not their turn.
Unlike MtG, where any player can use any card to construct their deck, in Hearthstone there are class-specific cards and neutral cards -- class specific cards can only be used for a hero of that class, and neutral cards can be used for any deck. This creates some interesting dynamics, as not everyone uses the same cards in their deck.
One of the more interesting aspects is that Hearthstone is a Free-to-play game; the portions that cost money are buying additional packs and buying entry into the Arena, but these can also be paid using in-game gold. The Arena is similar to Draft (which is my favorite style of MtG), except that it doesn't rely on other players being present for the card selection. This makes it so that you can make an arena deck and you aren't committed to a 3 hour long tournament as you are in MtG. Additionally, playing in the Arena grants gold, packs and materials to create cards.
The cards used in Arena don't belong to the player (so there is no need to pick for rarity), and any cards in your collection can be used for constructed play. The money required currently looks like this: Constructed Play is Free, and 3 Wins nets you 10 Gold. 100 Gold will buy you a pack, and 150 Gold will allow you entry into the Arena. The Daily Quest will grant anywhere from 40 to 60 gold (or more depending on what the objective is) This structure is what makes it possible for players to never spend a dime on Hearthstone.
The only game mode I actually care about is Arena, though, as without the rare cards to construct a solid deck, constructed feels a little too random, since one will likely be missing some very common utility cards. Since it takes forever to get enough dust to craft cards, the packs won through Arena play usually includes cards that will make your collection bigger overall, and more competitive over the long run.