The first pour, and what gets left in its top
On most foundation jobs, the footer goes in before anything else touches the ground. The trench gets dug and cleaned to the width and depth the plan calls for, the footer gets formed with board where the trench walls will not hold a clean edge, steel gets tied and held up off the dirt on chairs, and the footer is poured and left to cure before the wall above it ever gets formed.
A footer poured with nothing in its top is just a slab in a trench. Before the pour, steel runs continuous down the length, lapped at the splices and turned at the corners with bent bar instead of stopping short. And either a keyway or dowels get set into the wet concrete so the wall poured on top of it has something to lock into instead of just resting there.
Why it is wide instead of just thick
A wall by itself is a narrow line of concrete, and a narrow line concentrates load into a small footprint of soil. Widen that same load out over a footer twice or three times the wall's thickness and the pressure on any one point of dirt drops enough for the ground to carry it without settling. That is the entire job a footer does. It is a spreading device, not a structural wall in its own right.
Footer versus foundation wall, in one pour or two
The footer is the pad. The foundation wall is what stands on it and carries the building up to grade, usually poured a day or more later once the footer has cured. Not every job splits the two. A monolithic pour puts a thickened, reinforced edge and the slab in as a single piece instead of a separate wall on a separate footer, and which way your job goes depends on the building and the plan, not a fixed rule. Our foundations and footers page walks through that whole call in more depth.
The same pad, under a house, an addition, or a pole building
A footer under a house or an attached garage is a continuous strip running the whole perimeter and under any load-bearing wall inside it. Tie a footer into an existing house for an addition and the elevations have to line up so the two floors meet where they should, which means matching the existing footer's depth as much as the new plan. A pole building is the outlier. It usually sits on individual post footings instead of one continuous strip, since the posts, not a wall, are what is carrying the load down.