Microphone for Youtube Recording

Depends on how much you want to spend. You might want to consider a digital recorder having built-in mics like this one, which costs $150:

https://www.bhphotovideo.com/c/product/1453030-REG/tascam_dr_07x_stereo_handheld_digital_audio.html

I would suggest a mic or mics providing a "cardioid" pattern, which is the pattern provided by the two mics on that recorder, rather than a shotgun pattern which is more directional and may be too directional to capture the sound from two speakers in an optimal manner.

Also, if you go with a recorder such as that one it would be best to mount it on an inexpensive tripod, to minimize the extent to which it picks up reflections from adjacent surfaces. (And you would certainly want to avoid holding it in your hands while recording, as it would pick up handling noises). As you’ll see in the description it can be mounted on any of the many tripods which provide 1/4"-20 threads.

But if by any chance you might want to splurge on something really good, that would typically be used by pro audio people, I would suggest the following mic and mic amplifier/recorder, totaling $1250:

https://www.bhphotovideo.com/c/product/644381-REG/Audio_Technica_AT4051B_AT4051b_Cardioid_Condenser_Microphone.html

https://www.bhphotovideo.com/c/product/1333778-REG/sound_devices_mixpre_3_mixpre_3_mic_preamps.html

Generally speaking condenser microphones (as provided by both of the alternatives described above) are superior to dynamic microphones, although they usually cost more. And they must be used with mic amplifiers which provide "phantom power" (or with some separate source of phantom power), and which also tend to be relatively expensive compared to mic amps that would typically be used with dynamic mics.

Regards,
-- Al

First go watch some of the YouTube videos. There are two main approaches. The best one is to skip the mic and record direct from line. That might sound like a cheat but there's a whole series of these that are nothing but a video of a turntable and you listen to the actual line feed. Pretty easy to hear how much better it sounds than a CD, listening to a few of these even helped me pick my Koetsu cartridge and Herron phono stage. 

The other technique, recording with a mic, seems better but is awful. No room acoustics with any mic sounds anything but like you're in a barrel or something. Check out the Synergistic Research videos where they do this. They are GREAT for the comparisons Ted does. You can easily hear the difference between the system with and without their components. But while you hear the difference, what you hear doesn't sound anywhere near as good as the line feed videos.

So listen and see what I'm talking about. While you are at it maybe notice the mic setup Ted uses. He may even describe it in one of them. But realize recording like that if you want it to sound good is an art in itself. Which mic you use is the least of your worries.

Or you can use the technique used to record guitar amps. Put a Shure SM57 in front of your speaker, so 2 SM57, 2 stands, a field mixer, do the mix and equalization through uncoloured headphones and feed the camcorder/camera. You should be pretty close... :) The SM57 are cardioid mics with great response but this is not the only ones. After that you will have to suffer the YouTube compression that will ruin part of your hard work. So I don't think you can demonstrate anything this way.

Thanks for all the suggestions, I am going to look into these options and do some more research.
I have a new Integrated Amp arriving tomorrow and might want to do a review on it.