Disconnect the fountain from its power source. This will ensure that the pump is turned off and safe to remove for cleaning. Once the power is turned off, remove the pump and put it in the sink.
Empty the fountain. Follow the manufacturer's instructions for this step, or remove the water with a shop vacuum.
Use a mild soap or all-purpose cleaner and a nylon scrubbing brush or other non-abrasive sponge to clean the inside of the fountain.
Wipe away the soap and dirty water with wet rags, or suck it up with the shop vacuum, then wipe clean the inside of the fountain.
Remove the pump's motor cover to access its interior. Clean the inside of the pump, any holes and other crevices with a toothbrush. Wipe all accessible surfaces clean with a wet cloth.
Replace the cover and the pump, fill the fountain and reconnect the power source.