When you leave a fan on in an empty room, it would be smart to turn it off to conserve energy. This is because fans don't cool down rooms, but they cool people from the breeze it generates.
In order to properly efficently utlilize your ceiling fan in tandum with your air conditioner for energy savings. Only use your fan when the air conditioner is either off or when the cooling is turned down ten or more degrees. Using both at the same time won't save energy but will use more energy.
In the winter when you use your furnace, the hot air will rise to the ceiling, while the cold air will settle toward the floor. Reversing to fan's direction will make it pull air instead of push it. This effect will circulate the warmer air in the room down the walls and to the floor, keeping the room temperature at a more steady level. It will also put less strain on your furnace as well.
To further save energy make sure your fan is efficient. Energy Star has a minimum airflow standard of 1,250 CFM* and 155 CFM/W** at low setting and 5,000 CFM and 75 CFM/W at the fan's high setting.
*Cubic Feet per Minute
**CFM per Watt