Is there a website that is equivalent to the mobile app store but with webapp and website instead? Something like a library of websites where we can rate and discover new, interesting, and useful websites.

Getapp

Someone might want to stab me, but Chrome Web store is decent, and Android app support is coming soon I believe.

@magis No, seems reasonable, I was gonna suggest the Firefox app store, but I suspect that's long gone now...

Stumbleupon is great for discovering new websites. You tell out what you're interested in and it gives you a random website with every click of the stumble button. It will continue to update your interests with each like/dislike.