Inset - the door sits flush within the frame. More expensive.

http://www.dreamlinecabinets.com/wp-content/uploads/2013/03/photo-7.jpg

Overlay - the door sits on top of the frame. Partially or fully.

http://st.houzz.com/fimgs/7de24fb201e6f011_9236-w600-h376-b0-p0--home-design.jpg

Partial Inset - part of the door is inset into the frame but there is also a lip that sits over it.

I had old cabinets with partial inset previously. I don't think I saw any when looking, mostly overlay, which I am guessing makes manufacturing cheaper and easier. I have overlay now. The only problem I have with that is that there is as much as a 1/4" gap between the door and the frame so bugs have easy access (found a fly in there!). Those gaps don't keep the dust out either. I would prefer a fully closed cabinet.

I have inset. See how the door fits in the cabinet frame? It was more expensive when we did our kitchen. But, this time around, we wanted full overlay and that was more expensive than inset from one cabinet company by about $5K! Other estimate, inset was more expensive by about $3K. We are going with inset again this time. Here is my current inset kitchen:

Here is a pic of our kitchen in our Florida house. It has full overlay cabinets.