On an inset mount, the door has a rabbet cut in it all the way around so that the door sits with half of the thickness of the door to sit back and the other half overlaying the cabinet. It has sometimes been referred to as a lipped door.
Here is a quick look at a typical rabbet cut or joint.
In most cases you will need a 3/8” inset semi-concealed hinge. Part of this hinge will be mounted to the face frame and part of the hinge will be hidden from view behind the cabinet door.
There is another type of inset hinge called a "wraparound". This type of hinge offers extra strength by wrapping around the face frame. There are two types of wrap around hinges: full wrap around and half wrap around. Full wrap around full inset hinges function exactly like a butt hinge. The only difference is the "wraparounds" which give added support to both the door and the cabinet mounting.
You can see on the side of the hinge at the left that the hinge wraps around all three sides of the face frame.
And here is another style of wraparound hinge installed for a better idea of how yours might look.
We carry a decent selection of the most popular sizes and finishes of wraparound hinges. Many styles and finishes in wraparound are not very popular so we have a hard time justifying stocking them as a business matter.