The starter solenoid on a lawn mower acts like an electronic switch. When the ignition key for the mower is turned to the "start" position, the solenoid engages and sends electrical current from the battery to the starter. When the solenoid fails, there is no way to repair it, and it must be replaced.
Place the mower on a level surface. Open the hood of the mower and remove the negative battery cable by unscrewing the the cable nut with your ratchet and the correct socket.
Trace the positive battery cable to the solenoid. The first place the positive battery cable connects to leading away from the battery is the solenoid.
Disconnect or unplug each wire connected to the solenoid. In some cases, they will just be wiring harness plugs that can be pulled apart. In other cases, connecting nuts on electrical posts must be loosened before the wires can be taken off. Mark each wire removed by wrapping a piece of masking tape around it, and using a pen, label the post it came off. For example, a wire removed from the upper left post can be labeled "TL" for "top left."
Unscrew the bolts that anchor the starter solenoid to the lawn mower chassis. Take out the old solenoid.
Bolt in the new solenoid exactly the way the old one came out. Attach the wires to their correct electrical posts as marked on the masking tape, reconnect the negative battery cable and close the hood.