Magic Duels is a free-to-play, online version of Magic: The Gathering. I've played with it a bit in the past, and recently re-installed it to see what's new with the Shadows Over Innistrad expansion.
Magic Online, which does much more of that. Not every card from every set makes it into Duels, and there aren't big organized tournaments and other events. Which makes it good for casual play, without spending large sums on cards.
I play Magic Duels on my PC via Steam. It's also available on iOS and the XBox One. It's pretty much the same interface on all platforms, which means a limited UI by PC standards. Play consists of clicking on the cards you want to play, and pressing a button to pause play if you'd like to respond to something. That "pause to respond" interface mostly works well, although it's annoying at times while you wait for the pause interval to pass, or are a little slow and miss the interval when you wanted to pause.
With the release of each set, Magic Duels releases a set of campaign scenarios that players can complete, for gold rewards and some basic cards for their collection. These campaigns related in a general way to the storyline for the set. They're mostly easy to complete, although some will require a few tries to get the right draw.
When not playing the campaign, you can build decks and play against either the AI or other players. You can build decks either using a guided construction process for certain "archetype" decks, or a traditional deck builder. Since many of the daily quests require you to use archetype decks, I usually use that method, and play against the AI (generally on medium difficulty). Battling other players has the usual online-TCG problems: you see the same few powerful deck types repeatedly, meaning you have to play one of those yourself (if you have the cards) or lose repeatedly. I tend to avoid that mode for the most part.
The business model is more or less the same as other free-to-play TCGs. A new account gets a starter set of cards, and you can buy more with gold, the in-game currency. You get gold during play for game wins, daily quest completion, campaign completion, etc. It's a fairly slow process, so you can spend real money for gold to speed things up. There's also a "community quest" renewed on a weekly basis, where everyone who plays gets a gold reward when some goal is met across all players. That's a nice touch, something I haven't seen in other similar games.
There's one significant drawback to Magic Duels: bugs. Actually playing a match has been mostly bug-free (for me, at least), but issues crop up in other aspects of the game. Some examples: When I have a quest to win games with Blue/Black or Blue/White archetype decks, the Blue/White wins count, but the Blue/Black wins don't. Occasionally the game client will lock up when attempting to edit archetype decks (Green/Black seems particularly bad). After an update is released, my old decks will often have problems; a old mono-red deck that I tried to use ended up with no land, and nothing I tried in the deck builder would let me fix it (eventually I deleted it). None of this stuff is game-breaking, but it's very annoying.
Magic Duels is a fun option when I feel like playing a quick card game. It definitely has its problems, though: limited UI, restricted card availability, and bugs. Works as a quick time-waster, but I don't recommend spending any significant time (or money) on it.