In addition to an input voltage range what are the important characteristics of a three-terminal regulator?

When selecting a three-terminal regulator (like the popular 78xx series) for a circuit, the most critical design parameters are the specific output voltage it is designed to provide and the maximum output current it can safely deliver without overheating or failing. While input voltage range is also necessary to ensure the regulator has enough overhead (dropout voltage) without exceeding its maximum rating, 'minimum' output currents or voltages are generally not primary limiting factors for these devices. Manufacturers specify them primarily by their fixed output voltage and their maximum thermal/current limit (e.g., 5V at 1A).