What do these passive components do in this microphone amplifier circuit?
R5 and R6 set the gain of the circuit.  

C4 technically changes the frequency response, giving it a LOWER gain at higher frequencies, but the value of the cap implies that it's just there to add some stability to the circuit and won't effect the audio frequency response too much.

C2 also changes the frequency response of the circuit-- reducing the gain at low frequencies.  Basically, cutting off the bass frequencies.  I am assuming (I have not calculated) that this will cut off frequencies below about 200 or 300 Hz.

Technically, C2 is doing the same function (High Pass Filtering) as C1+R2+R3.  Odds are high that you could remove C2 (short it out) and have a circuit that works just fine.

UPDATE:  I should also mention that this circuit is missing a DC blocking cap on the output.  This is normally done, especially when the opamp is ran off of a single supply rail.