In American English a broiler has heat above (gas or electric), when you put meat in it you are broiling it. A grill has heat below (gas or charcoal). When you put meat on a grill, you are grilling it. In both cases, the meat is heated on one side only, so you must turn it over to cook both sides.
In British English, we do not distinguish heating from above or below, both are called 'grilling'. The term 'broiling' isn't used much nowadays, it refers to cooking meat in an enclosed pot or pan with some fat in the bottom. We still sometimes refer to an old chicken as a 'broiling fowl' because traditionally an old chicken would be cooked in this way, it would be too tough and dry to roast.
An oven is a closed, heated box, if you put meat inside the box to cook, you are roasting it. The heat is even on all sides of the meat.