WARNING: The Binding of Isaac leans very heavily on gross-out imagery and the occasional ancient meme for humor. A lot of people find the game enjoyable in spite of that, but for others the constant barrage of piss, blood, and feces is too much. So...don't say we didn't warn you.
Okay so a few years ago, Edmund McMillen and Florian Himsl made a quick and dirty flash game as McMillen wanted a fun thing to work on after the arduous development that was Super Meat Boy (note: Himsl is not part of Team Meat and this is not a Team Meat game!). So in about two months the duo released the Binding of Isaac, and then a month or so later came the Wrath of the Lamb expansion because McMillen's wife loved the game and wanted more content. To be very quick, the game is basically like the dungeons from the first Legend of Zelda. The difference is that the game has permadeath, random layouts, and a ton of random items you can find to make your character grossly overpowered. As you play you unlock new levels, items, and characters to keep things interesting. The game was a slow burn, taking a bit to really get popular, but it eventually took off like a rocket. People plunged hundreds of hours into it, and they wanted more.
The problem was that BoI was made in Flash, and they were already reaching the upper limits of how much they could add to the game (although this year Himsl did release a patch/mini-expansion he'd been working on) and there were other issues that kept the game from improving like native controller support. So McMillen got together with the people at Nicalis, and they worked with Himsl to translate his code and give the game a proper release. The result was the Binding of Isaac: Rebirth, a combination of remake and expansion pack that gave the game a superior engine with improved graphics, controller support, and a lot of new bells and whistles like tons of new content. Releasing last year, the game was definitely a hit with fans, providing even more hours of dungeon crawling weirdness.
It didn't take long for McMillen to announce an expansion, and that is Afterbirth. Besides standard features like new characters, items, and enemies, the game also features a whole new way to play via Greed Mode. This is basically like a Horde Mode where you fight waves of enemies followed by a boss wave with breaks to buy items. The game also adds daily runs, which are seeds that give you one shot to play for the highest score (the game features a scoring system now, by the way), although you can do more runs after you die that just don't go up on the leaderboard. There's also apparently more stuff still locked away, and Edmund's playing some kind of game as to how to unlock it, so that's something else to be excited for, I guess.
Rebirth is basically everywhere: PC/Mac, Xbox One, PS4, Vita, New 3DS (NOT on other 3DS systems), and Wii U. Afterbirth is currently only available for the PC/Mac, but it's coming to other hardware starting with PS4/Vita at some point later this year. The PC or Mac is probably your best bet because it gets everything first and you can also mod the game to extend the lifespan or just cheat for unlocks if you don't wanna do goddamn Lost runs.
Quick and Dirty FAQ
What happened to the music?
McMillen and DannyB have stopped working together, so Rebirth got a new soundtrack. If you really miss the old soundtrack, then here is a guide to change it.
What does [X] do?
This site will be your cheat sheet for all items.
How do I unlock-
Here you go.
Is there online co-op?
Nope, still local only.
What do the symbols under my collectibles mean?
Crossed-Out Trophy: You can't earn achievements during this run (seeded run or most challenges)
DR: Means you're on a Daily Run.
Red Pentagram: Hard Mode
Shoe: This Daily Run or Challenge ends at the Mom fight.
Heart: This Daily Run or Challenge ends at the Mom's Heart or It Lives fight.
Sword: This Daily Run or Challenge ends at the Satan Fight.
Cross: This Daily Run or Challenge ends at the Isaac fight.
Filled Box: This Daily Run or Challenge ends at the ??? fight.
Empty Box: This Daily Run or Challenge ends at the Lamb Fight.
Brimstone Icon: This Daily Run or Challenge ends at the Mega Satan fight.
What happened to my saves in Afterbirth!?
Look at the bottom of the screen for an import option. Note that things are a bit buggy and you will probably get a bunch of achievements you may have no legitimately earned. This is to be fixed sometime this week, hopefully.