The stove is the whole appliance while the oven is just the compartment with the door. A stove usually has 4 burners and 1 oven. The burners are on top and are usually round.

Usually the word stove refers to the entire cooking device which includes the oven and the stovetop. The stovetop (also called the cooktop or the range) is the part with the burners. The oven is the part that you bake with. Sometimes these are separate devices, in which case the word stove would only refer to the stovetop/burners, and the word oven would only refer to the thing you bake things in. We say "on the stove/stovetop" and "in the oven" because of the different places we put the food when using one or the other. Note that I'm writing this from California. I believe different vocabulary is used in other areas.

I'm just adding to what others have already said. I'm from England and we often call the stove a cooker (stove is rather old-fashioned, though we use it for wood-burning stoves in the lounge which are for heating). Yep, I think cooker and oven are the most common terms these days.