With an unregulated power supply, the output voltage will vary depending on both input voltage and load. Even if the input voltage is fixed, the output voltage will be higher if lightly loaded or lower if heavily loaded. An unregulated power supplies would be a transformer, bridge rectifier or capacitor.
With a regulated power supply, there is an additional voltage regulator in either linear or switch-mode. Voltage regulators use a feedback loop to constantly adjust voltage drop or duty cycle – for linear and switch modes, respectively – so that the output voltage will be constant regardless of load and input voltage.