What is called the extension of the nervous system that is in our skin and, when we touch...

something too hot (a pan...), it sends an impulse back to the brain saying our skin (hand) is being burned?

The peripheral nervous system refers to all receptors and nerves outside of the brain and spinal cord - that's the overall name for the system as a whole. Within the peripheral nervous system are various types of receptor which code for different stimulus and are experienced as different sensations.
A good wikipedia page can be found here: http://en.wikipedia.org/wiki/Somatosensory#Periphery


For pain - Noiciceptors sense pain and the quality of sensing pain is called nociception.

When your hand is burnt you may trigger a reflex action. Some reflex actions do not even need the inpulse to go all the way to the brain - it can go to the spine and trigger a movement response directly from spinal cells.