Shut down the power to the outside light. If you think that the problem might be with the light switch, go to your home's breaker box and turn off the breaker controlling the light. The electrician that installed it should have labeled your breaker box.
Remove the light bulb from the spotlight. Turn the bulb counter-clockwise to remove it from the light. Put your new bulb into the light socket and then turn it clockwise to install it.
Go back to the breaker box and turn the power back on. Go to the switch that controls the light and turn it on. If the light will not turn on, you should check the wiring going to the light. Go back to the breaker box and turn the power off.
Remove the light cover from the spotlight. Screws usually hold this on. Look over the wiring inside the unit casing. Make sure all wires are connected to the light socket. If any of the wires have come loose, use a soldering iron to reconnect them to the light socket. Your light should have come with a wiring schematic to help you determine where the wiring should go.
Turn the power back on once you have reconnected the wiring. Turn the light switch on. If the light still doesn't come on, go back to the breaker box and turn the power back off.
Remove the light switch cover from the switch. Look for loose wires on the inside of the switch. If you find any loose wires, replace the switch with a new one. If you do not feel comfortable replacing the switch, call an electrician to help you out.