Emacs can run as a server, and you can connect multiple local clients to it. I've tried various ways to have an emacs client connect to a remote emacs server (forwarding a socket over ssh, etc.) but never gotten it to work so there must be more to it than just the socket.
But it does run in terminal mode - I used to ssh into a remote machine and just run emacs in a terminal there. Actually, there was also some `screen' in the mix, but you get the idea. I preferred that over TRAMP because of the speed.
No I don't get the idea. I was disabusing people of the widely believed myth that an emacs server instance could host remote connections. That one can ssh into a remote machine and run emacs in tty mode is manifestly obvious.