My furnace will come on and run normally and heat to whatever temp my thermostat is set at. It will then cycle off as normal. Whenever it starts up again that's when the trouble starts. The furnace begins to go through all of its "pre-cycle" steps it normally goes through:
1. Exhaust fan kicks on and runs for 10-15 secs
2. Pilot light kicks on and glows for another 10 secs (pilot light is element, not gas burning)
3. Gas kicks on and I can see all 3 burners light up - these will burn for another 10 secs or so before the blower fan kicks on.
(I'll break up step 4 to what NORMALLY happens during cycle 1 vs what happens for all subsequent cyles)
4. NORMAL - blower fan kicks on, pilot light turns off, burners stay on, furnace works normally.
4. WHAT HAPPENS NOW - blower fan kicks on, burners turn OFF immediately, and fan blows air for a few minutes until system turns itself off.

At this point I can see the circuit board inside the unit and there are 2 LED lights that blink during normal operation: RED light is on always, and there is a YELLOW light that blinks on and off during normal cycle start-up, and then stays on during normal operation.

However now after the furnace tries a few times to start itself back up (remember the thermostat is still sending signals to the furnace to light up) it will eventually shut down altogether and the RED light will blink 4 times in a row, over and over again as if it were sending an error message.

I do not have the manual to this furnace so I do not know what those 4 RED-light-blinks mean.

There is a power switch located outside the furnace that I have to switch OFF and then back ON to clear this error message (Red light now stays on as if in standby mode). I can then turn the thermostat back on and the furnace will run normally for ONE CYCLE ONLY. After that, well, read above!

Is the systerm over-heating?? If it were then why does it operate normally after I manually flip the power switch and it works fine for one cycle? Does this sound like something electrical? Are the burners being turned off because of some safety feature?

Ideas??

Frigidaire, model FG6RA O72C-16B.

Try cleaning the flame sensing element. It may be that the sensor gets hot during the one cycle and degrades its milivolt signal to the ignition module so that the next time it starts and can not see the proper milivolt signal. Once you turn it off and back on you reset the lock out and the sensor has cooled allowing it to gain signal for one more cycle. Flame sensors are inexpensive and easy to replace.