A "mom and pop" store refers to a small family store rather than a larger corporation. Often this means that there will be only one of these stores and it is managed directly by its owners. A grocery is simply a store that sells mostly food products to be used elsewhere. This could be vegetables, meats, etc. Also a grocery can also be a mom and pop store. Note that a grocery is different from a restaurant in that a restaurant you buy food that is prepared and cooked for you to eat and in a grocery this is usually not true.

This is probably an american phrase, I've never heard of it in Britain. We would probably refer to this kind of shop as "the corner shop" and it does not matter if the shop is on a corner or not.