How do I open this FPC connector?
I've encountered this sort of connector a few times. What you want to do is pull upwards on the grey flap on the top of the image (though it could also be the bottom half that flips up, I've seen both ways) and once that's up, it releases the spring pressure on the flat flex and lets it slide out easily.

I'm not sure if there's a name for this sort of connector, but they're common, at least in my experience, in laptops. I think they tend to have a lower profile than other sorts of flex connector, and they are much easier to insert the flex into than friction-held connectors.