This is Rogue Legacy, an indie roguelike side-scroller in which you slash your way through randomly-generated dungeons to fight monsters and earn treasure. Here's the big gimmick: if you die, you get to play as your character's child, and then when you die again, you'll play as that character's child, and so forth. Every generation inherits loot from the one before them.
Each child has certain traits: a kid can be colorblind, for example, or hyperactive. If you're OCD, you'll gain hit points by breaking every pot you see. Naturally.
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Sounds neat, yeah? Rogue Legacy is out today for PC and Mac. You can check out the demo on their website.