Why stove top and why oven?
The stove top provides high heat for searing, which generates browning (the famous maillard reaction) which creates delicious, savory flavor components.
The oven provides an even, gentle heat for longer cooking without burning.
The beauty of the braise
Braising is a technique where a food, usually a meat, is cooked slowly in a moist environment, partially covered with water. It is an ideal technique for tougher, generally less expensive meats, because it converts collagen to gelatin, changing the meat from tough to meltingly tender and wonderful.
Braising is more common than you might think. Chili con carne, and pot roast, osso bucco and coque au vin are all braised recipes.
Searing and braising
Many braising recipes start with a sear, which is most commonly done stove top, because of the high intense heat that is easy to generate. Alternate methods would be to use a broiler or an outdoor grill, but these are often not as convenient. The purpose is to generate the savory flavors (not as the old kitchen myth said, "to seal in juices").
Then the dish is finished in its liquid (which will later usually become the sauce) at a much more gentle cooking pace.
Braising is flexible
You can braise in the oven. The advantage is the heat is very even and predictable, and you can just let the pot sit in the oven for 2, 3, or even 4 hours without worrying about burning or sticking.
You can braise on the stove top, with a gentle flame (or lower electric element setting). You simply adjust to leave the pot contents at a slow simmer (bubbles form lazily, not continuously). There is slighty more risk of burning, and the occassional stir might not hurt, but especially with a good pan, no risk.
So almost any braising recipe can be easily converted to be stovetop only, by simply... erm... using the stove top for the long braising time. I prefer the oven personally, but that is a matter of convenience and easy control.
Quality cookware like good multi-ply stainless steel, or your now famous dutch oven (from your previous questions) is ideal for braising because it can be used for both the sear (just like a cast iron pan) and for the slow braising portion. No need to dirty two pots or pans.
Recipes that are just stovetop
There are a countless myriad recipes that you can cook in your cast iron just on the stove top.
- Deep fried
The one thing you may choose to avoid doing is highly acidic recipes, like tomato sauce, which are not great for your seasoning. On the other hand, in rotation with other recipes that rebuild the seasoning, or just doing some direct seasoning, there is no true harm done either.
Recipes that are just the oven
There are many recipes that would only be done in the oven.
Many braises skip the searing part. Even recipes that call for it can be done without, at some loss of flavor.
Any recipe you see for a crockpot can be done in a dutch oven in the regular oven.