To find it definitively, you'd need to:
* Consult your owner's manual: This is the best place to start. The manual will have diagrams showing the location of major components, including the ECM.
* Look for a large black box with wiring harnesses: Systematically search the area near the firewall on the driver's side of the engine compartment.
* Use a repair manual specific to your 2003 Buick Century: A Haynes or Chilton manual will have detailed diagrams and explanations.
Caution: Be aware of hot engine components and use caution when working near the engine compartment.