Assuming that you mean headset (if you mean headphones you will require a microphone), first I would check that the headset is working on a different device. In my experience a lot of cheap headsets break very easily. If it doesn't work on another device then it is definitely going to be a hardware problem with the headset.
Next you need to check that the headset is compatible. The connector should fit snugly into the jack. If you have a single jack headset (like for a smartphone) and you are trying to plug it into a computer which is expecting separate audio output and microphone inputs then that is not going to work. If you only really care about making the audio recording it might work if you plug it into the microphone jack but I can't guarantee that. Also if you want to listen back to the recording you'll either need to keep switching between the audio output and microphone jack or get a separate audio output device.
So let's say that everything checks out you have a microphone plugged into the microphone jack (and perhaps audio output plugged into the audio output jack) and they fit properly.
Then I think they next question would be to check that you are up to date:
Try following these instructions:
https://wiki.ubuntu.com/Audio/UpgradingAlsa/DKMS
If it still isn't working try a restart.
If it still isn't working try this in the terminal:
sudo apt-get install alsa-tools-gui
hdajackretask
And try a restart. After that I am out of ideas (without further input).
Hope that helps!