On will cause the fan to run continuously. Usually not necessary. But there may be times when you want fan only without the AC compressor running.

Pretty sure it's on auto Fred. I can't explain it, as I just follow directions on this stuff.

Ken Jones is right.

Also, if you have and AC/Heat combo unit, and you call for heat, the furnace usually turns on for a few minutes, then the fan kicks on when all is up to temperature.  You don't want to blow cold air through the house.

Mine is set to auto so that it only runs when the AC turns on. 

Otherwise, if set to ON it always runs. 

Y'all are welcome to refer back to my post;  When Should You Get Your A/C Ready for Summer?

A number if issues that need to be known. 
But first, let me state in FLORIDA the fan should be set to RUN not auto. Air movement is exacly why most closets have the vented doors.
BUT, heres the problem. If the ceiling as lost integrigty (meaning there are pathways from the airconditioned part of the house to the unconditioned part)  the fan must be set to auto. Continuous RUN will suck the conditioned are out faster through the integrity breeches.
It is important in a FLORIDA environment to make certian wall outlets, ceiling lights, attic access panels, stud based air returns and undersized ductwork do not violate ceiling entegrity or system efficiency. And there still exists in Florda those who love to throw the windows opned and spring us turn on the attic/ceiling fan. Thise fan installation methods are the equivelint of a open window in the summer. No ceiling integrity.
Constant air motion means  the termostat can be set to 78 comfortabllly. Or, like is most of florida, a flock of low effecieicny paddle fans serve the same purpose.

"Auto" is the way to go unless the air outside is the right temp for you and "on" is appropriate. A rare happening for most.

It depends, do you want to circulate air all the time or just when the air conditioning or heat is on?

According to my A/C guy, AUTO.  If the fan runs all the time, especially if the unit is in an attic, it can overheat.

According to my husband, the home is cooler if it is ON all the time.

The fan is primarily used to circulate air in the house through the filter and thus the unit.  Good to use if you have an allergen in the house, or burnt the popcorn (guilty).  Another good use case is if you want to open windows and force the inside air out (such as hotter inside than outside).  "On" means it will always be on and running.  Mine also has "Circ" which will run it up to 40 minutes of every hour.

Auto will ensure that your fan turns on when the AC or heater is active.  Without that, the cold or hot air would have no where to go.

well, this is what I have been told. It's easier on the motor, etc if the fan stays in the ON position most of the time.  I don't like it except when there are many people in the house like a party.

I think it is noisy being continually on.

Auto here, and we only have heating...no need for AC with ocean breezes...A

Fred, I believe Ken has answered your question.  On mild spring, summer, or fall days when the temperatures are between 60-85 we will turn the system off and run the fan as our house sits on the north-south exposure and is well shaded in the front to the north and west so we keep plenty cool with just the fan on and AC/heat off,  if the temperature stays below 85 degrees.  We keep cool doing this and save money and reduce energy costs.  If the fan is set to on and you have the AC on it will run continuously.

In the winter the full windows on the back of our house will warm the great room up to 72 degrees and it is toasty warm so we turn the thermosat down to 66 and the heat doesn't kick on till about 8 or 9 pm.

Auto is what I believe is recommended.

Mine are set on "auto" my thinking is once the temp reaches my desired setting it will turn off a while then back on as needed to keep home comfortable. 

auto, why would you want to move air ALL the time and wear your fan motor out by never giving it a break.  In winter when heating with the fireplace we turn the fan "on" for short periods to move the air around.

Is this a trick question? Auto with thermostat set to whatever. 

LOL this is more complicated with evap cooling . . . gotta turn it on mid morning, then off before bed or you will wake up to a VERY cold house. 

Once it's AC season (another month or so) evap 'summerized' house closed up and AC at 78 . . . if you have it 'ON' all the time it will constantly be running even with temperature control,  waste electricity and tax the system. 

Reporting live from the surface of the Sun . . .