This is a standard overlay door. You can see how the door doesn't cover the faceframe completely. This is an old-fashioned design that's not at all popular anymore. Do not buy standard overlay doors! The arched top rail on this door pushes this style closer to the dustbin where it belongs.
This is a full-overlay framed cabinet door. See how the door and drawer front cover the face frame? This is what you want, regardless of the door style. The door style I'm showing here is what I call a wide-stile Shaker door. It's the more modern take on a typical Shaker door.
Finally, there's inset cabinetry. Rather than resting on the surface of the faceframe, inset cabinetry doors sit inside the faceframe. Generally, inset cabinet doors are made with either a beaded or unbeaded frame, and with either visible or hidden hinges. The image above shows a non-beaded cabinet frame with hidden hinges on the door.
Here's a beaded frame on a cabinet with hidden hinges. The bead in a beaded frame is the outline you can see around this door. That bead is an added detail and serves no real purpose but to make the door appear to be larger. Beaded inset cabinetry is usually available with either exposed or hidden hinges.
And here's a non-beaded frame with exposed hinges. When you're looking at inset cabinetry with exposed hinges, there are usually three or four hinge finishes available. In the image above, the hinges are finished in a matte black and they have a ball finial. An exposed hinge can give a modern-ish door like this one a more vintage feel.
Inset cabinetry is a traditional form of cabinet making and it's been making a comeback in recent years. Although it's a traditional form, it doesn't have to be used in a traditional setting.
The kitchen above is a transitional contemporary design with some traditional elements --the Hoosier-type cabinet between the wall ovens and the well-disguised refrigerator jumps out at me as the big traditional element here. That Hoosier cabinet is actually made from inset cabinetry by the way, and all of the cabinetry shown in that image is by and available from Medallion Cabinetry. If you'd like a good run-through of a solid and well-made inset cabinetry line, check out Medallion's Platinum line.