Please edit, if 'engines' is not the correct term (layman here).
As I understand it, you have your SRB's, then they detached, and the External Tank supplied the liquid oxygen and hydrogen fuel to the main engines of the Orbiter. Then, after main engine cutoff, it is detached.
Why didn't the External Tank just have its own engines/boosters? The Orbiter maintained a bit of fuel for orbital maneuvering, but otherwise didn't need to be responsible for the thrust provided by the ET.