Hi Steven,
Your circuit is exactly the same as the output section of my Electronic Stethoscope-2 project, and the same as on the LM386's datasheet. The only difference is that it doesn't have enough gain.
Connect a 10uF cap between pins 1 and 8 of the LM386 with its positive wire at pin 1.
I connected an electret microphone to a circuit like that and could hear my voice in headphones very loudly with the microphone at arm's reach and the volume turned down. I could even hear a TV that was in another room. Your circuit has a gain of only 20, and adding the cap boosts the gain to 200, at which point the LM386 has a noticeable hiss output. Without the cap, a low noise preamp would boost the gain without hiss. A dish feeding the mic would make it much more sensitive. Use a rubber isolating sleeve around the mic.
Please disregard what Kevin said (sorry, Kevin):
1) The mic must be an electret type (not a coil and magnet type) and of course must be connected with the correct polarity. An electret mic is very common , has high quality and is inexpensive. It has a much higher signal level output than a coil and magnet mic.
2) R1 is used to power the FET transistor inside the mic. Its current may damage a coil and magnet mic.
3) The LM386 power amp (it is not an opamp) operates normally with both inputs at ground voltage and has built-in biasing. With its single supply, the output voltage will be near half supply voltage. That is why it has an output cap to the speaker.
4) The output of an amplifier (not an LM386) would be zero volts if it had a dual supply.