The first mike is in a headset designed for computers
That should work as long as you have the correct connection. Sometimes they have separate connectors for the headphones & mic, and sometimes there is a combination 4-conductor TRRS connector. If the connections on the computer & headset don't match, you need an adapter cable.
the second is a studio type mike (both are pictured).
Stage/studio mics are
not compatible with regular consumer soundcards or the mic input on a laptop, and computer mics are
not compatible with mixers, PA systems, or mic preamps, or USB audio interfaces.
Stage/studio mics have a balanced (3-wire) low impedance connection. The two signal wires are out-of-phase, which helps to cancel noise/hum pick-up.* The differential input on the mic preamp amplifies the difference signal and kills the common signal (which is noise-only).
The mic input on a computer is not a differential amplifier and it's not designed for stage/studio mics.If you use an adapter and split one signal to the left and the other to the right, they will be out-of-phase. If you just kill one channel and make mono, you
might get acceptable results.
If you combine those two signals together to make mono, everything cancels. If you listen to the out-of-phase stereo you'll get some "phase weirdness" as some of the sound waves mix and cancel in the air.
Studio condenser mics also require 48V phantom power (supplied by the preamp, mixer, or interface). A computer can only supply 5V for an electret condenser mic. Dynamic mics (like the famous Shure SM57/58) don't require any power. There are some stage electret condenser mics with an internal battery.
I'm pretty sure you have an electret condenser mic that designed to look and work like a studio mic. (I don't know if it can work properly from 5V.) A real studio condenser mic usually costs $100 USD or more.
* It's only for electrical noise pickup... The balanced connection doesn't help if your dog starts barking in the middle of your recording session.
