The battery
Solenoid
The side of it.
starter part were the wires are.
If the mower has an electric starter, the gear on the starter is worn. Replacement is not hard and the part (a plastic gear) is relatively inexpensive.
The starter solenoid is attached to and an integral part of the starter.
The starter motor
That is part of the starter and is mounted on the top of the starter.
It is part of the starter.
The solenoid is part of the starter.
If the lights in the car come on, then at least part of the system is getting power. Check the battery connections again. Then check the starter solenoid and starter to insure that they are getting power and are working properly.
It is part of the starter. It has a clip for the starter wire labeled s.
The solenoid is built on the starter.