Auto

The fan starts when the system starts and stops shortly after it finishes. In cooling, this lets water on the coil drain away between cycles, which helps the system control humidity.

On

The fan keeps moving air through the house and the filter even when nothing is being heated or cooled. That can smooth out warm and cool spots, but it uses more electricity, and in humid weather it can blow moisture from a wet coil back into the house.

Other settings

Some thermostats offer a circulate setting that runs the fan for part of each hour. It is a middle ground between the two.

A common mix-up

With the fan set to on, air from the vents feels warm or room temperature between cooling cycles. That can look like the air conditioner has stopped working when it has not.