Stumps can be massive in a big tree. A large tree will have a stump with many large roots growing out into the soil. You can cut the surface wood away, but the entire woody root system remains. As wood decays in large stumps and roots, the soil may subside, or holes may develop in your landscape. Sometimes animals use these subterranean homes.
Stumps are safety and aesthetic concerns.
Traditionally, removing a stump has meant digging and cutting, then repeating, until you or the stump gives up. Some have used explosive charges to free the stump from the soil. Other people try to burn the stump out. Still others use an archeological approach, leaving the stump as an artifact. New equipment and services are available.