The battery is like a giant capacitor that helps the alternator supply the car's electronics and accessories with filtered DC current. Disconnecting the battery while the engine is still running will introduce pulsating DC to the electrical system. And since modern cars are highly dependent on a filtered supply of DC, it can affect the ECM/PCM logic, sensors, etc. Problems can range from being temporary (ie. the engine shuts down) to something permanent (ie. weakest link in the chain of semiconductors bites the dust).

The way I test it is to take the baseline voltage of the battery with the engine turned off and comparing it later with a sample battery voltage while the engine is running. Then further take voltage tests with the accessories and lights working. And finally, rev it a few times and see if the regulator is working by observing the output of the voltmeter.