What are Private Servers?
A private server is a server not hosted by SuperCell that you can connect to to play if you root/jailbreak your mobile device to edit the host file to allow you to access a different IP address with the game. You can get unlimited resources and gems on some private servers.
As for if they are legitimate, it depends on your definition, they exist, but are most likely not legal. I am not a lawyer, but Supercell's TOS seem to forbid this, specifically in section 1.1.