The starter relay receives the electrical signal sent by the ignition switch when you turn or flip the key, or when you press the start button. The relay contains a core that moves forward and pushes the fork, which in turn moves the drive arm, allowing it to turn the engine flywheel. The engine starts, and then the drive arm returns to its original position.
The starter relay is also called a solenoid or, more traditionally, a contactor.
Replacing this part may sometimes be enough to repair the starter; to find the correct model, you must always use the exact part number of the starter.