You can connect to the remote machine using ssh with X11 forwarding (-X) enabled. Then any graphical programs you run on the remote machine will have their output forwarded to the local display.
That can be condensed further if you only want to run one program and then exit afterwards.
ssh -X -l raincityrunner othermachine.example.org
You'll want a fast connection with low latency because it can be a little sluggish otherwise. Usually it is fine over the LAN.
ssh -X -l raincityrunner othermachine.example.org firefox