What is a private IP address? In which scenarios/system designs, one should use it?

Private IP addresses are assigned to the hosts in the same network to communicate with one another. As the name "private" suggests, the devices having the private IP addresses assigned can't be reached by the devices from any external network. For example, if I am living in a hostel and I want my hostel mates to join the game server I have hosted, I will ask them to join via my server's private IP address, since the network is local to the hostel.