What are the functions of these capacitors in this circuit?
Your first circuit uses an IRF2110 MOSFET driver.

The capacitors you refer are there to smooth the voltage that the  IRF2110 generates to drive the FET gates.  The IRF2110 is used to make a higher voltage in order to drive the gates of the FETs turn the FET fully on.  FETs require a certain voltage differnce between gate and source in order to switch fully.  If your circuit can't provide the needed voltage, you use something like the IRF2110 to generate it using the available voltage.

There are two capacitors because capacitors aren't perfect.  If they were, a single large capacitor would be all you need.  Because of imperfections (large capacitors act a bit like an inductor at high frequencies) you use a large capacitor to catch the low frequency noise and a smaller capacitor to catch the higher frequencies.

In your second circuit, the capacitor you marked functions as a filter.  It works together with the 1.2M resistors (two in series, so 2.4M total) to form a low pass filter with a cutoff frequency of around 30Hz.  Any frequency above 30Hz will be reduced in amplitude.

I don' know why you need that filter with that cutoff there because the rest of the circuit is missing (it would help if you had explained where you got the circuits.)