Yes, they get their power from the board. Any amplifier board that has a built-in electret mic, or is designed to work with an external electret mic, should work with your headset.
Ideally you should separate the supply circuit (just a resistor coming from the positive supply) from the signal path (a series capacitor, feeding into an amplifier circuit) and bring the power source to the ring and connect the signal coupling capacitor to the tip, so that your board would also work safely with other types of microphones that don't need to be (and shouldn't be) powered from the preamp.
The diagram above shows how a condenser microphone is connected on your preamp/amplifier boards. There is a resistor, usually around 4.7 kΩ, from a positive supply rail to the electret, which provides power for the internal circuitry inside the microphone, and there is a coupling capacitor that passes the signal from the microphone to the rest of the preamp circuitry. Electret microphones are designed to use a single connection for positive supply input and signal output. The other connection, which is connected to the metal case of the microphone, connects to the 0V rail (aka ground).
The diagram above shows how an electret condenser microphone (for example one in a handsfree kit) connects to the microphone input of a PC or sound card. The external microphone is on the left, and the circuitry inside the computer or preamp is on the right. This circuitry now has the power feed (through the resistor) and the signal feed (through the capacitor) separated onto different terminals on the socket. Specifically the supply is provided on the ring, and the signal is taken from the tip.
If the external microphone is an electret condenser type, and needs a supply, then it is wired as shown, with the tip and the ring connected together. This completes the circuit between the resistor and the capacitor, so the result is like the first diagram.
This is how an external microphone works when it doesn't need (and doesn't want) power from the computer or preamp. The microphone is connected only to the tip of the plug; the ring terminal of the plug is not connected ("NC"). The positive supply through the resistor is not used in this configuration.
If you want to modify your preamp to use a jack socket instead of an onboard electret condenser microphone, I recommend that you wire it like the two examples above. Then you can use it with either type of microphone without potentially damaging a sensitive, high-quality microphone that isn't designed to have a supply voltage applied to it.
The difference between a preamplifier and an amplifier is the signal levels they use. A preamplifier is designed to amplify a low signal level, from a microphone, turntable pickup, etc, up to "line level" - something in the region of 0.5V RMS, although there are many different standards. A line level signal is not able to drive a loudspeaker; it may be able to drive a sensitive headphone but is not designed to do that either.
An amplifier takes a signal at line level and amplifies it up to drive a speaker. You can also get headphone amplifiers, which don't amplify the signal level much but provide a low impedance output suitable for delivering some power to the headphones. (Line-level signals are medium-impedance and cannot provide much power.)
I've just seen your recent post. Here are some comments on the kits you found.
Combines a preamp and an amplifier, effectively. Can drive a small speaker, or headphones. I don't like the LM386; there are much better options available now, but this is usable. You should modify it as I described above to use a jack socket.
Ditto. This one has RCA phono sockets for input and output.
Ditto, except this one doesn't include the resistor to supply the electret condenser microphone. That might be an advantage, since it means no changes to the circuit when you add the jack socket; just add a resistor from the positive rail to the ring terminal on the socket.
This one uses a type of amplifier that's designed for "bridge-tied load" (BTL) operation. These are not well-suited to driving headphones.