[kaptain]

This looks like an electronic logic board that controls when the 12v air compressor turns on or off.
The 4x black 3-leg secured by hex-screws on top of the board, marked as ST TIP32 are 4 power transistors, https://www.st.com/resource/en/datasheet/tip32c.pdf
The multi-legged IC marked as P9336 is a binary counter, which my guess is that it compares the signals from the pressure sensors, so it turns on/off which power transistor (thus kicking on/kicking off the compressor).
The fact that it uses 4 power transistors instead of just 2, it probably has reverse-compression also, i.e. When pressure is too high to a dangerous value that is stored in EPROM, it activates two of the 4 transistors so actually takes air away from your system (via activating the relief valve or by reverse rotating the compressor).
The rectangular component marked "Program 51" is likely an EPROM that stores certain values that the manufacturer set for trigger values. It's not a complicated controller, and likely it will not produce an error code or display when encounters fault.
The burn marks are no big deal, it's just normal aging when boards get older, especially for resisters that generate heat. 
If you suspect the burn mark caused some damage, just use a multimeter to measure the resister according to its marking. You can find on the Internet what the color code rings corresponding to the supposed to be resister value.