I believe what you are referring to is an acceptance flight, rather than a certification flight. Certification flights (usually there many) are flights which the manufacturer demonstrates to aviation authorities that a new design of an aircraft is safe. However, just because a design has been tested, does not mean a new aircraft is produced exactly to the design. Acceptance flight is a test flight which make sure the aircraft has been manufactured up to standard.
Acceptance flights are also required after major maintenance work or avionics upgrades. Its purpose is to explore the entire flight envelope and ensure emergency / backup systems work as expected. During a normal flight, only a portion of the systems' functions are used. A test flight is a complete evaluation of all conditions, such that malfunctions or deviations will not go unnoticed.
Of course, the pilots will first examine critical systems (e.g. engines, flight controls) on the ground. Depending on the aircraft, the tests to be completed in the air include:
- Engine shutdown and restarts
- autopilot roll rates, flap extension speed, engine response etc.
- Alternate gear extension
- Fuel jettison
- Over-speed warnings
- Stall protection systems
- Ground Proximity Warning System (GPWS)
- Auxiliary Power Unit (APU) operation at high altitude
- Cabin service systems