Shut off the existing pump and unplug its power cord, unless it's a new fountain installation without a pump in place.
Open the drain for the fountain to empty the water. The drain varies with the type of fountain. Large outdoor fountains might have a dedicated drain and lever system that opens to divert water through a pipe away from buildings. Smaller fountains might use a drain plug inside the basin.
Lift off the pedestal in the center of the basin to expose the pump compartment underneath.
Loosen the screw in the metal or plastic band surrounding the hose on the pump. Pull off the hose.
Remove any mounting hardware holding the current pump to the basin, using a screwdriver. Raise the pump to break the seal on the suction cups underneath.
Pull out the old pump and the power cord from the gasket at the bottom of the basin.
Thread the power cord on the new pump through the rubber gasket in the basin and pull the cord from below the fountain. Extend the cord to the power outlet, but do not plug it in yet.
Seat the pump in its compartment and press firmly to seal the suction cups to the fountain basin.
Adjust the pump speed to the desired level. This is typically a knob control on the pump.
Attach the hose to the port on the top of the pump and secure it by tightening the screw in the metal or plastic band.
Replace the fountain pedestal over the pump compartment.
Fill the fountain with water supplied from a garden hose.
Plug in the power cord.