The stump does not come down with the tree. A standard removal cuts the trunk as close to the ground as practical, but it leaves the stump and the root system in place, because dealing with them is a separate task that calls for different equipment. What you do about that leftover stump is a decision you get to make, and there are a few distinct paths.
The most common choice is stump grinding. A machine with a rotating cutting wheel chews the stump down into wood chips, typically to several inches below ground level, so you can cover the spot with soil and grass. Grinding does not pull out the entire root system, but it removes the visible stump and enough of the upper roots that the area becomes usable lawn again. In Georgia this service usually runs $130 to $430, with the price tracking stump diameter, and many homeowners have it done the same day as the removal while the crew and equipment are already on site.
Full stump removal is the more aggressive option. Rather than grinding the stump down, the crew excavates the stump and the major roots out of the ground entirely. This leaves a larger hole to backfill and costs more because it is more invasive and labor-intensive, but it is sometimes necessary if you plan to build, pour concrete, or plant a new tree in the exact same spot.
Some homeowners simply leave the stump. There is nothing wrong with this if it is out of the way, though a stump will slowly decay over years, can sprout new growth from some species, and may attract insects as it rots.
If the stump sits where you mow, walk, or want to landscape, grinding is usually the practical middle ground: affordable, quick, and enough to reclaim the space without the cost and disruption of digging the whole root mass out.