At first this feels like a burn, then like someone's putting needles into your hands, and then they just go numb. You can't do precise actions with your hand anymore and soon you'll lose most of the ability to move it at all. You might even lose the body part. All while the core of your body is still warm and you're still able to walk and talk.
But as said, not everyone experiences this. For some people, when they get cold, their body increases circulation in the hands, keeping them warm enough to continue working no matter what.
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1. In extreme cases, this is called white hands syndrome or reynauds syndrome and primarily affects women. It seems to have a hereditary component, but worsens permanently whenever the hands experience cold or vibration.
