I've never heard of this series. What is Zero Escape?
Zero Escape actually started as 999: 9 Hours, 9 Persons, 9 Doors for the DS. There is an iOS version of the game, however it is generally considered the inferior version for reasons I will go into momentarily.
The gameplay of 999 is a mix of a visual novel and "escape the room" puzzles. These puzzles are similar to what you would find in a point and click adventure game, similar to something like Monkey Island or King's Quest. The visual novel sections of the game are narrated and contain dialogue between the characters in the game. There is some interaction from the player during these sections, and some of the choices really do make an impact on how the game will play out. Most of it though is filling in the storyline though.
The iOS version of 999 is considered inferior to the DS version for a few reasons. The main reason is that the puzzle rooms are missing, and while the story is still there, you are still missing out on the game's world by being able to interact with the puzzle sections of the game. There is also one other reason, however I can't go into specifics on it since it is major spoilers for the game. All I will say is that you won't truly "get" a major plot element of the game's storyline unless you play it on the DS.
The game had a sequel, Zero Escape: Virtue's Last Reward. VLR is available on both the 3DS and Vita. In this case, play VLR on whichever system you prefer or own. There are differences, such as some bugs with the 3DS version, the Vita version having higher resolution, and the note taking functionality is different in the two versions, but other than that, you will get the same experience out of it. Some people may say the 3DS version is awful, but I played it on 3DS myself and it was just fine.
Before VLR released, Aksys Games, the publisher of the games in the series in the US, rebranded the series as "Zero Escape". The reason for this was that the series needed some kind of name that made it clear there was a connection between the games. The name "Zero Escape" is chosen because all the games in the series involve an antagonist that goes by the name "Zero" and escaping some kind of area.
Visual novels are a genre that is a hard sell on many people, and understandably so. Most people don't want to play video games to read, and that is an understandable concern. The genre overall isn't necessarily for everyone, however I think the Zero Escape series is for everyone if you have a bit of patience and engage yourself into the game's world. You'll find that the Zero Escape series does a superb job of playing with your emotions, expectations, and mind. Most of the gameplay in the ZE series is mental, encouraging you to think out of the box, and even then shattering your expectations.
In both games in the series, you will find characters that are all interesting, clever puzzles, and one of the best told stories in video games. You may have thought a game like Telltale's The Walking Dead, or The Last of Us was at the top of the line in storytelling, but they don't hold a candle to this series. The games I mentioned aren't bad games, but they definitely are not as good as the Zero Escape series.
At this point you are probably wondering why you should even bother trying to get through the walls of text. Giving a justified answer to that question is a very difficult one without getting into spoilers, so I will say, like many others in this thread, and any other Zero Escape related topic, to just trust us on this one and play the games for yourself. I would also recommend not reading more of this thread before playing the games, as much of it is spoiler-filled theorycrafting for what will happen in the third game. You can find 999 on Amazon, however at the moment it might be backordered. You can usually get it for around $20 though when it is available. As for VLR, you can get that game physical on 3DS or Vita, or digital via Nintendo eShop or the Playstation Store.
Why all the hype for Zero Escape 3?
Again, I will try to give a justified answer to this question without getting into spoilers.
VLR ended on a huge cliffhanger, as well as a confirmation from the developer that the series would be a trilogy and end on the third game in the series. A while back, the director of the game, Kotaro Uchikoshi, said that since VLR did not sell well in Japan, that Zero Escape 3 would not be made. This news disappointed many fans, and there was a huge outpouring of support on Twitter, including even a fan campaign to bring Zero Escape 3 back from the abyss of cancellation. After Saturday's panel, we learned that this caused the higher-ups at Spike Chunsoft to reconsider the decision, and Aksys Games stepped in to help revive the game. The game sold very well in North America and Europe, and given that this was where the audience for the series was, the game was able to be revived with help from Aksys. It almost seems like a miracle the game exists, but it does, and we are all thankful for it. We also got other news that the tone of ZE3 would be more like 999, which had a much more macabre horror tone than VLR. VLR was definitely not light-hearted, however it did contain some pandering, and toning down of violence to try to appeal to the Japanese market. This did not work, and the game sold even worse than 999 did.
Conclusion
I know it's hard to take some people's word for it, especially on a visual novel, but just trust us and play these games. You will be glad you did. Before I played 999, I was put off by the idea of visual novels, but I had heard such high recommendations for 999 when it released, that I decided to pick it up, and now I can say it is easily one of the best experiences I have ever had in video games. I hope you pick up the game and experience this same feeling yourself.