There are two main parts of a chimney—the chimney stack and the chimney breast. The chimney stack is the part of the chimney that extends above the roof—the part visible from outside the home. The chimney breast is the part of the chimney that is visible from within the home.
Partial and complete chimney removal both involve removing the chimney stack, but only complete chimney removal involves removing the chimney breast too. Both methods have their pros and cons, so you will have to compare them to determine which removal method is best for you.