Can anyone tell me the difference between the auto and fan/on options
on a thermostat that is connected to a gas furnace.  My roommates are
arguing about the best way to save on the gas bill and this has come
up.  Does either setting affect the rate at which gas is used?  I was
under the impression that auto will cause the furnace to kick on when
the temperature dips below the set level and fan recirculates the air
in the house--using electricity, not gas.  Any help that I can use to
stop my roommates from squabbling with each other would be great!

In the "auto" position, the fan that circulates the air in the ducts
will run only when the furnace is burning gas for heat.  In the "On"
or "Fan" position, the fan that recirculates the air runs all the
time.  The fan runs on electricity.  If you are worried about only the
gas bill, then running the fan will not raise the gas bill.

I have a theory that running the fan all the time will use less gas in
the long run.

In the "auto" position, there is a delay between the time the gas
ignites and the fan starts to blow.  The purpose of this is to allow
the furnace to heat up so that it does not blow cold air when it first
starts.  But during this time, at least some of the heat produced is
wasted.  If the fan blows all the time, then all the heat will be sent
out to the house.  Likewise, when the house reaches the set
temperature, the fan and the gas fire will stop.  There is additional
heat in the furnace that can be blown out into the house if the fan
were kept running.
Also, with a continuously blowing fan, the air is moved around the
house, evening out the hot and cold spots.

However, some people like to find the hot spots, and don't want them
disturbed by colder air blowing them around.

Thanks!  That's exactly what I wanted to know.  If you'd like to
resubmit it as an answer (I'm not sure what exactly that involves, I'm
new at this), I would be happy to give you due credit for the info.

Ohbonobogirl,

Philnj is not an official Google answer person (their online ID is in
blue), so you don't need to pay him, nor can you.  He just made a
helpful comment because he wanted to.