A pool pump is driven by a motor that starts under power from a capacitor. A failed capacitor will cause the motor to hum or buzz when the start cycle is activated. The buzzing noise will not be accompanied by rotation sounds from the impeller of the pump. In that case the pump is not working and the pool water is stagnant. Two types of pool pumps are available, and both are started by capacitors. Test the capacitor to see if it's all that's needed to restore service to a pool pump.
Shut off the power to the pump with the breaker switch in the nearby breaker box.
Open the pump motor cover. The round cover is at the end of the pump cylinder. Use a slotted screwdriver or 1/4-inch socket wrench to remove the two cover screws. The capacitor is the cylindrical object held in place by a curved metal guard plate.
Drain all stored power from the capacitor. Touch both terminals on the top side simultaneously with the metal post of an insulated screwdriver, which will cause a spark and popping sound if the capacitor has a lot of stored power.
Remove either of the wires from a terminal on the capacitor. Use needle nose pliers to grip the connector, and unplug it from the terminal.
Test the capacitor with an analog multimeter if it is rated at 150 microfarads or less. Higher rated capacitors can ruin analog meters. Set the meter for "Ohms" on a 1k scale. Touch the meter leads to separate terninals, and watch the needle on the display. Either lead can go on either terminal for the test. The capacitor is bad if the needle drops to zero and stays there. The capacitor is good if the needle jumps toward zero and slowly swings up the scale to show greater resistance.
Test any capacitor with a digital multimeter that has a capacitance setting. Set the meter for "Capacitance" and touch the leads to the terminals. Either lead will work on either terminal. Check the digital display to see if the number comes close to the rating shown on the capacitor. If the numbers are off by more than 10 points, the capacitor is bad.