A ceiling fan does not cool the room itself; it moves air and helps people feel cooler. It saves energy only if that comfort lets you raise the air-conditioner setpoint or run cooling less.
The effect is on people, not room temperature
Moving air increases evaporation and improves comfort. A thermometer may show nearly the same room temperature with the fan running.
Pair it with a slightly higher thermostat setting
If you stay comfortable with the air conditioner set a little warmer, the compressor runs less. That is where the real savings come from.
Turn it off when the room is empty
Because the main benefit is how moving air feels on skin, leaving it on in an empty room usually wastes electricity without adding comfort.
Use seasonal direction settings only if supported
Some fans can reverse direction to mix room air in winter. Use low speed and follow the manual; do not expect the fan to replace heating or cooling.