How do I fix a motor that hums but won't spin?

Start with the start capacitor — that little cylinder in the back of the motor. Give the shaft a quick spin with a screwdriver (power off, then briefly on); if it starts running, the capacitor is dead. Replace it — cheap and easy, part number on the side. If it still hums with a nudge, you’ve got a seized bearing or dead winding — just buy a new pump motor.