In the first game, you play a bunch of guys (Lightning among them, but despite being the first character and on the cover of the box, she's not really the main character) that have been charged by ineffable AI machine gods to do some vague task. It ends up being that they're trying to use you to break out of their programming that forces them to maintain humanity. If they don't complete their task, they become zombies, if they do complete their task, they turn to crystal until the machines decide to let them go.
In the second game, Lightning was turned to crystal, but she somehow gets recruited to protect time, and is off fighting this huge war, and so she recruits some more people to jump around to different times and places trying to correct things that have been thrown off because of her big war. Not really sure what happens next, but things apparently get fucked.
Third game starts off, and Lightning wakes up from her crystal stasis, and is charged by God (maybe the guy that made the machine gods in the first game?) to help prepare for the next world, because this one is ending. God's going to wake up in 13 days, and when he does, the world's going to end and be rebuilt. Shit's fucked though, and the world's on track to end in 5 days, in which case it can't be reborn. Lightning has two tasks: Keep the world in one piece until God shows up, and collect souls from people so that they can be reborn in the new world. You do this by completing quests where you give people hope, which preps their souls and gives you special energy that you literally infuse into Yggdrasil which extends the lifespan of the world. Also, apparently time broke at the end of the second game, and no one's aged for 500 years. So even though Lightning's been asleep the whole time, everyone she fought with in the first game is still kicking, but also, there are 500 year old children all over the place.
It's apparently like Majora's Mask where you run around doing stuff, and people will only be around at certain times for you to complete quests, and then the world will end, then you start back at the beginning with all of your stats and gear, but all of your quest progress is reset. Things get even weirder with the gameplay, though. You don't have levels in this one, which isn't that odd, but you also don't have a skill tree. Your stats go up when you complete a quest. Finding this guy's cat might get you 80 more hp and a point of magic, getting medicine for that kid's mom gets you 40 hp, and two points of attack. Outside of that, everything is gear based. You have four ability slots, corresponding to the four face buttons, and the outfit you're wearing usually dictates at least one of them, and then you can equip any ability you want in the empty slots.
All three games have heavily featured this idea of swapping between multiple loadouts in a fight. In the first two, you were just switching between different classes, basically, all of your healing stuff was on one class, and you'd have to switch to that class to do any healing. In the third game you only play as Lightning, and rather than having explicitly defined classes, instead you can switch between three gear load outs for her that you equip with different abilities and different weapons. The weirdest part of this is that you can only perform so many actions you can perform in a row, but each set of gear has it's own gauge for this, so you can blast through all of your actions on your first set of gear, switch to the next and use all of its actions, and then do the third, by the time you make it back to the first, it's ready to do a full round of actions again.
It's just bizarre, but in a way that's refreshing. I haven't really read any review of FFXV, so I've no idea if it continues in the same vein of all this weirdness, or if it's gone to something more comprehensible. I did watch the prequel movie, so I've got a bit of a feel for the story, and while it seems pretty out there, it's not like this.