What is this circuit board and how to I replace it?
Of course it is impossible to tell for sure without looking at the circuit in more detail but it looks like a 120V AC to low voltage DC power supply. 
A way to troubleshoot it would be the following:


look up the voltage rating on the lights. Using a multimeter, check if this voltage is present at the output of the board. If it is, the problem is with the lights/contacts, etc.
check for a blown fuse on the board (if present). If you find a blown fuse, get a couple of new ones and replace it. It might have been blown in some sort of a fluke but if it keeps breaking there is a problem with a circuit. If no fuse:
Switch the meter to AC Voltage measurement and measure the output of the transformer (large part with green core). If there is no voltage, the transformer is bad. If there is, you would have to redraw the schematic and troubleshoot every component (caps are usually the suspect). At this point it might be cheaper to get a new unit.


As with any high voltage board be very careful when troubleshooting a powered board!

PS: I believe Monso is wrong here - the voltages after step down transformer are typically 5-24 V which is not enough to cause harm (after the board is powered down).