I thought that looked like a Bryant control board before you took the cover off..lol.. good pics. That relay will not unplug, so you will have to replace the board.
I found a exact replacement board, well sorta of. Its the replacement board as close as you are going to get.
Only biggest difference i can see right off, is the extra terminals for the A/C, which on the current one you have now, there is none.
So you can actually hook this one up the same way the one you have now is.
RED= 24V...WHITE= Heat...Green= Fan for A/C or Manual Setting @ thermostat.(which is HIGH speed),Bigger Black wire on control board going to blower motor.
Bigger Red wire you have now on control board, is LOW speed, which is for heat.
2 wires going to transformer which is LINE Voltage(120V) to power transformer. Other 2 wires leaving transformer is the LOAD side, which is 24V going back to your control board.
Not too hard really. Think of it like this: The control board is a Motel that holds all the switching and working of the system.
The control board first has to have 120V supplied to it. Then from there, the control board supplies 120V to Transformer, which then the transformer makes 24V, and plugs back into control board to work all your 24V controls.
Your thermostat is fed with 24V from the red terminal on the control board. The 24V @ your stat, is just sitting there, waiting for YOU to tell it what to do. For example,
You click stat to Heat, and then the 24V is moved from the red wire, to the White wire on your stat, which is sent back to the control board. Once the (W) terminal (which happens to have a white wire) is energized on the control board with 24V, it starts the heating cycle. As long as (W) Terminal is energized, the heating cycle will stay on, or call for heat.
It then sends 24V through all your limits, and if there "closed" as they should be, it completes the circuit and the relay for the spark ignition will Begin sparking, and pilot light will light, or hot surface ignition will glow, depending on your application.
Once it lights, then the thermocouple(or flame sensor if hot surface) will let the gas valve open and ignite the burners. in 120 seconds(rule of thumb), the control board will energize the blower relay(close), and give 120V to the RED wire and the fan will begin running.
Once (W) terminal is de-energized due to you turning off the stat, or the stat satisfying due to temp setting, the gas valve will close, and the relay will continue to be closed and let the fan run about 180 seconds(rule of thumb), then open the 120V it was supping to the red wire, thus stopping the blower..... In a nutshell.
Knowing this, you should be ok to change out the control board. Here is a link from carrier itself i found.
Furnace Control Circuit Board Replaces two Clear Relays.(BDP, Bryant, Carrier, Payne): American HVAC PartsLet me know what you think, or what else i can do to help ya.
Also, do me a favor. Bryant, Carrier were notorious for having cracked chambers. So before you spend money, remove the cover over the 3 burners, and get a mirror and a flashlight and look up into each chamber,(which you have 3), and make sure you see no cracks, or gashes in them. here is a couple of examples: