A country ham is one that's dry-cured over a long period of time, producing a proscuitto-like product that's very dry to start with. It generally comes uncooked.Typical cooking method for American country ham may involve soaking it to remove some of the salt, then baking or roasting in a sauce, or frying. Some people will boil an entire country ham, and then fry slices to crisp them up a bit.
A "city ham," or as my family used to call them, "packing house ham," is one that's not cured nearly as long and is often either hot-smoked or otherwise cooked before selling. That's the typical spiral-sliced hams or other packed-in-plastic hams you find in the grocery. Most, but not all, have water or some such solution injected into them, and in any case, they're a much more moist product.
Country ham is a much stronger flavor, and it's quite tough if not cooked for a long time in a wet medium of some sort. There's a definite funk to it, as well as the strong saltiness. City hams are the ones that are most often glazed with a sweet glaze. Deli ham is a close cousin of city ham.
I have to confess that, while hearing about Smithfield hams often, I have never had one and don't know where they fall on that spectrum.
You didn't ask, but I will tell you the Appleton Farms spiral sliced half-ham from Aldi is one of the best city hams you can get, at least as far as I'm concerned. I order my country ham in pre-sliced packages, either breakfast steaks or biscuit pieces, from Broadbent's in Kentucky.
