Live linux is a linux distribution that runs from a DVD, USB or other removable drive and which does not change things on your hard drive. It is like a virgin system that makes no permanent changes to your computer.
An air gapped (not airtight) computer is one that is never connected to any computer network. The term dates back to when we connected to computer networks using physical wires i.e. ethernet cables. To be air gapped means that their is gap between the cable and the port on the back of your PC i.e. the cable is not plugged in so there is no connection to the network.
In this day and age an air gapped computer would not be capable of network access via both wired and wireless means. So it would not have a WIFI card or a mobile network access. The hardware for such wireless access has to be removed from the computer for it to be a true air gapped PC (because otherwise deeply embedded malware could be making such connections without your knowledge).
The idea behind an air gapped computer is that any malware that might infect the computer can't communicate with the outside world so it can't steal your secrets i.e. your private keys.