I suggest checking and ruling out issues one by one.
1. Firstly check to see if your mic is working properly. Use it on another device like a mobile phone or another PC to see if it works. If it works, move on to the second step.
2. Secondly, check to see if your PC does not have an issue with the microphone port. Also check the microphone settings under Volume. If its mute or disabled, enable it. Also check the 'Device Manager' and see if you have proper drivers installed for audio and microphone. If all's well, move to the third step.
3. Connect the mic to the PC and log onto Skype. Go to Tools - Options.
Under Options, click the Audio Settings tab on the left. Now talk using the microphone and see if the green bar under the Microphone volume moves up and down. If it does, then everything seems right. If it does not, move on to the next step.
4. It may be possible that you may have more than one microphone attached to the PC. There maybe one inbuilt and an external mic connected. Under the Advanced tab on the same screen, select the primary microphone. If this works, great. If it does not, move onto the next step.
5. Skype lets you make a free test Skype call. Use that. This call lets you check your audio and microphone settings. When prompted, record an audio sample using your mic and see if it is played back. If it still doesn't play back, only GOD can help you :-)
I have tried my best to think of all the possible problems and their solutions. I hope one of the above options solve your issue. Please revert back and do let us know how was it solved.