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Cheap mikes (electret condenser mikes) require that a supply voltage be fed into them through a resistor. The same pin is used as output. A capacitor helps to extract the AC output signal from there. An operational amplifier can be used to boost the voltage by, say, a factor of 100. Note that this will introduce noise and also boost unwanted signals such as powerline hum. By charging a capacitor via a diode and discharging it via a resitor, the envelope of the boosted signal can be extracted (somewhat inprecisely, but ok for basic applications). This envelope signal can, for instance, be fed into the Arduino.
Operational amplifiers produce negative and positive output voltages. Hence, they need both a positive and a negative power supply. The possess two voltage inputs (+ and -), of which they form the difference and amplify that difference by a factor of 1.000.000.000 or more. In typical applications, this huge gain factor is reduced strongly through a negative feedback such as R2 in the schematic. The schematic represents the most simple use of an operational amplifier: an inverting amplifier with gain of R2/R1 (actually, minus R2/R1, due to the inversion).