Final Fantasy XIII was released in December 2009 in Japan, and released in the rest of the world in March 2010. As of 2013, it's sold 6.6 million copies worldwide. It's playable on the Playstation 3 and Xbox 360 consoles, as well as the PC.
The game takes place on Cocoon, a floating, constructed world above the larger planet of Gran Pulse. Humans live on the interior of Cocoon, and are governed by a human body called the Sanctum. Magical beings called the fal'Cie maintain Cocoon, keeping it afloat and providing everything from light to food and water to its people. These fal'Cie are capable of branding individual humans and turning them into their servants, called l'Cie. L'Cie are given great magical abilities to fulfill the task the fal'Cie have given to them, called their Focus. If the l'Cie fails to complete their focus within their alloted amount of time, they will turn into a Cie'th, a shambling monster. If they succeed, they will turn into a crystal.
Yeah, it's still Final Fantasy after all. Gotta have crystals!
Upon its release, it was clear that FFXIII was by far the best-looking Final Fantasy; the technical advancements are leaps and bounds over anything Square Enix had achieved previously. Every single cutscene looks like the FMVs of old — and there are a lot of cutscenes, because FFXIII is plot-heavy, especially from the start. Each of the six main characters gets a lot of development over the course of the game.
The battle system is completely new for the Final Fantasy series, and while at first it seems like all you do is push the A button a lot, it's a lot more complicated than that. The paradigm system assigns each character a role in battle — for example, medic is healer, sentinel is tank, and commando is physical attacker. You don't have to worry about inputting every single attack, defend, and heal command; what you focus on is the overall role your characters are performing in battle. With a quick hit of a button, you can swap your characters' roles out so you've got a team of tanks, healers, mages, whatever's necessary at the time. It makes battles fun and fast-paced, and while you can still input individual commands if you want, it's generally easier to just let auto-battle take care of it while you focus on the bigger picture.
If you've played FFX, you'll find FFXIII very similar; it's linear up to a certain point, and then you're able to do some sidequests before continuing on with the plot. The level-up system, the Crystarium, is much like the sphere grid, albeit with fewer choices until they're unlocked about halfway through the game.
Like I mentioned above, you can pick up FFXIII for the PS3 and the 360. I've played it on both; the PS3 is slightly better visually, but it's honestly just a matter of which console you prefer to play it on. The 360 version comes on three discs; the PS3 version is just one.
The game has not been released digitally on the Playstation Network or Xbox Live; you'll need to purchase a physical copy if you're looking to play it on console. There's no DLC for the game; everything's on the disc.
As of October 2014, you can also play FFXIII on the PC via the Steam platform.