Zero Escape is a visual novel puzzle game series comprised of three games: Nine Hours, Nine Persons, Nine Doors, Virtue's Last Reward, and Zero Time Dilemma, released in 2009, 2012, and 2016, respectively.
The games are split into two modes, puzzle and story. During the puzzle section, the player must solve a puzzle in order to escape a locked room. In the story section, the player sometimes inputs a choice to direct the story, but otherwise watches things play out. All three games feature multiple endings, and in general, this is a series in which the point is to reach every ending in your pursuit of the true ending, rather than stopping at one ending.
The premise of each game in the series is similar: nine people are trapped in a facility and are forced to play a game in order to escape. The particulars differ quite a bit between each game in the series, but let's just put it this way: they're not playing checkers.
In general, the Zero Escape series is best experienced if you know as little as possible about it before playing, so if you're interested, I strongly recommend the series. It's full of fantastic plot twists and great storytelling; I couldn't put any of the games down.
As a note, the games should absolutely be played in release order, and yes, you have to play all three. Do not skip one of them or play them out of order. Trust me on this.
As of late March 2017 an updated rerelease of the first two games will be released on Playstation 4, Playstation Vita, and Steam under the title Zero Escape: The Nonary Games. Otherwise, 999 is available on Nintendo DS and iOS, VLR is on Nintendo 3DS and Playstation Vita, and ZTD is on Nintendo 3DS, Playstation Vita, and Steam.
999 is also available on iOS, however it does not include the puzzle segments, and is not fully compatible with current versions of iOS (in my experience). I actually recommend playing it on DS if at all possible, although I realize it's tricky to find these days.
Please note that if you get VLR on 3DS, it has a game-breaking bug if you save in certain escape rooms. I did play it on 3DS and avoided saving in any escape rooms and was fine, but do be aware of this.