What type of hinges do I need for this cupboard?
It sounds like you need a "concealed door hinge" This will let you close the doors flush with the surface and the hinge will not be visible when the doors are closed.

Here is a link to a hinge page with a few possibilities.

http://www.mcmaster.com/#standard-cabinet-hinges/=onl7hs

We order from McMaster Carr regularly. They are a reputable company.

They also have magnetic push to open latches, so you don't have to have any hardware at all like handles showing either.

http://www.mcmaster.com/#magnetic-latches/=onlapj

Good luck on your project!