Attached is a simple non-inverting amplifier based on an op amp.
Your application - the amplification of a signal of fairly high level from an electret microphone - doesn't need a really high performance op amp, so the lowly LM741 should do fine, as would the aforementioned TL071 or TL081. The pin-outs shown are for the single-section op-amps mentioned, but dual-section op amps could be used if the appropriate pins are used for a single section - you'd just leave the unused pins floating - which wouldn't matter in this application.
How it works:
R1 and R2 are equal resistance and form a "virtual ground" at "mid-supply" voltage with C2 filtering out any noise and residual audio. Because the non-inverting input needs a voltage reference, R3 biases it at the same mid-supply - but since there is a little bit of audio that gets back to R1/R2, C2 removes that audio.
R4 and R5 work together to form the feedback circuit to set the gain, with the left side of R4 working against our "ground" (again, C2 is important here as it sets the impedance low at audio frequencies) and the voltage gain of this circuit is:
1 + R5/R4
You can read more about this circuit here:
https://en.wikipedia.org/wiki/Operational_amplifier#Non-inverting_amplifier
The values aren't critical at all, except as noted:
R1 = R2 - They must be the same, but they may be anything from 1k to 100k. I'd pick 10k.
R3 - This should be pretty high to prevent loading of the electret microphone so anything from 10k to 100k would be fine. I'd pick 22k-47k.
R4 - This partially depends on the potentiometer that I have around, but you don't want it to be too low. A good range would be 1/20th of the maximum value of R5 or so - or even a higher ratio - let's say, 10k.
R5 - If I had a 100k potentiometer, this would give me a maximum voltage gain of 11 (about 21dB) and if I had a 500k potentiometer this would give me a maximum voltage gain of 51 (about 34dB). It all depends on what you need. I suspect that the 100k will work in your case - but you can always decrease R4 OR increase R5.
R6 - This sets the bias on the electret microphone and the value depends on the microphone itself. If it's an unknown microphone I'd start out with 10k if V+ is 9 volts: You should see 1-5 volts at the junction of R6/C1 when the microphone is operating normally.
C1 - This would effectively set your low frequency response. If you need, say, 100 Hz as the low end, you can get away with something as low as 0.1uF and be just fine - but I'd not go much above 10 uF. (A polarized is shown, but it not need be.)
C2 - This should be large enough in value to bypass audio that might appear. A value of 10uF-100uF is fine - I'd probably use 22uf-47uF.
C3 - This sets the coupling of the output. If it's driving a high-impedance output, 1uF-10-uF is fine and if it's lower impedance, you might need to go as high as 47uF-100uF. I suspect that your application would be OK with something in the 1uF-10uF range.
C4 - This bypasses the power supply. Often, you can get away without this since the batter that one might use has enough bulk capacitance to do the job. I always put at least a 0.1uF there for good measure, but a single 10uF would be fine, particularly if the power source was located some distance away from this circuit as the lead length can cause instability.
V+ - This depends partly on the op amp that you choose. "Officially", the '741 and the TL071, etc. may not be happy with a partially-discharged 9 volt battery, but usually all that happens is that the output gets anemic rather quickly when the voltage drops down. Since are not likely to be needing the full output of the op amp under any circumstances, I suspect that a 9 volt battery would be fine, but anything up to 18-24 volts (assuming that the capacitors' voltage ratings are adequate!) would be suitable for the '741 or the TL071/TL081. If you choose a different op amp, you'll have to check its ratings!
One thing of which one should be aware: With an op amp like the '741, don't expect it to work really well with a voltage gain much above 20-50 or so with a good frequency response - but that's probably not much of a concern in your application.
Best of luck!