A doorway is the place through which you enter a room. Some doorways have actual doors, while others are merely arched openings into a room or building.
If you stand in your doorway watching your neighbor approach on his bike, you've probably got the front door open and you're standing in its space. The doorway to your living room, on the other hand, might be a simple arch or wide entryway. The word dates from about 1799, from door and way, which here means "space" — it's the space in a wall where a door sits.