OK firstly on pretty much all computers that I have ever seen you do not have seperate headphone and speaker sockets... they are the same socket and can be used for either... sometimes it is refered to as a speaker socket and sometimes as a headphone socket but it's the same thing.
Your microphone plugs into the microphone socket, which can often also be used as, and is sometimes called a line in socket, but again this is the same socket.
pugging a microphone into a speaker socket will not work... ones an input the others an output!
I would suggest that you double check what socket is what on your system and make sure you have the correct thing plugged into the correct socket.
if, when you are sure that they are, the microphone still doesn't work then double check there is not a mute switch on the headset/microphone somewhere that you have knocked by mistake also double check that you do not have the mic input muted by double clicking on the speaker icon down by the clock and making sure the mute box below the microphone volume slider hasn't got a tick in it.
if it still won't work then it may be a broken lead on the microphone and you'll need to replace it