With how many options your thermostat can have, it can be confusing to know what settings are right for you. Many homeowners are especially unsure if the "on" or "auto" fan setting should be used. To a large degree it depends on your preferences of energy savings vs. comfort.
The Fan Doesn't Heat or Cool
The first thing you want to know is that your fan is separate from the heating and cooling settings on your thermostat. It will always run when there is a call for heating or cooling. However, by setting it to "on" you can also have it circulate regardless of whether the thermostat is in a heat, cool, or off mode. For example, it can be nice to have the fan running when it's a comfortable temperature in the house and you just don't want the air to feel stagnant.
Energy Savings & Comfort
If you are concerned with how operating your fan affects energy usage, you will either want to set your fan to "auto" or have a variable speed motor (electrically commutated motor or ECM) . When a conventional fan motor (permanent split capacitor or PSC) is always "on" it uses more energy because it will run on high speed continuously. Set on "auto" the fan only turns on when the unit needs to heat or cool the space, cooling typically runs on high speed and heating runs on medium or low.
ECM motors are programmed to operate at 20 - 35% of their normal airflow (depending on the manufacturer) when there is no call for heating or cooling, which gives constant circulation without operating at high speed. Plus, an ECM uses approximately 1/3 of the energy of a PSC motor even at high speed.
If comfort is your main concern you will want to set your fan "on" to keep the air moving and reduce hot and cold spots. With the fan "on" it will also keep the air moving through your filter and bring in a small amount of fresh air, this type of operation also provides better indoor air quality (IAQ) than the "auto" setting. Keep in mind you may need to change your filter more often as it will be filtering air 24/7.
Variable Speed Motor
With an ECM you will have energy savings and more comfort. The energy savings come from the motor having a very efficient direct current (DC) magnetic pulse, the control module in the motor also provides precise airflow because it can self correct for moderately dirty filters or a few closed registers. The motor also speeds up gradually which keeps it quiet when it turns on.
So, What Should I Do?
If you have a conventional fan motor choose "auto" to lower energy costs, or choose "on" to balance temperature between rooms and improve IAQ. To get the best of both worlds use a unit with an ECM.
If you are replacing your furnace consider an ECM (blower motor) to maximize comfort and minimize energy consumption. The upfront investment can be higher, however you generally recover the cost through energy savings, are more comfortable, and improve IAQ.