Well, I have to spoil half of the story with this but still...
The game starts when you save a girl's life, and it turns it's Chloe, she used to be your your best friend before you moved out then came back. So you get to be friends again, especially since she is looking for another one of her friends, who is missing and whom she spent a lot of time with since her father died and you were gone.
Since you can time travel, at some point you get to the point when her father was about to leave the house just before dying, and you undo that. And when you get back to the present, Chloe is paralyzed after a car accident and only has days to live before she dies. After you spent one day with her, she tells you that day was the best day in her life and she wants it to be her last memory, so you can just unplug her (more like "put the level of medicine to the highest", but same idea). First sad death.
Of course, you don't like that so after basically killing her, you undo what you've undone and you get back to the timeline where her father is dead and she is alive and in good health. But after some more adventures (basically figuring out what happened to her other friend), it turns out that death really wants Chloe since you saved her several times, and there is a tornado that is going to wipe your town. So your only solution to save the town is to get back in time to the first time when you saved Chloe, at the beginning of the game, and let her die, so she never knows all you've done. From her point of view, her father died, her other friend disappeared and she doesn't know what happened to her, and you were gone and never came back. Second sad death, the worst one. That's exactly my theme on my New 3DS, if you recognized it, Chloe and you looking at the tornado and figuring out what's to be done.
The worst thing is that these are choices, you don't have to kill Chloe. You can refuse to unplug her or/and to save the town. But you know you have to. You can't let her suffer and you can't let a whole town die.
