Sounds like you need either X forwarding or VNC.
X forwarding allows you(via SSH or some such) to execute a program on the remote computer but have all the display "commands" be sent to your local computer. It's notoriously slow, but is the best solution when you're not dealing with graphically intense thingsAND you're just wanting to see a single window and not the whole screen (and your LAN is fairly fast, ie 100Mbit or more). X forwarding also lets you control the program and such with a mouse from your remote machine.
VNC on the other hand will basically take a series of screen captures on the remote machine and forward them over to your machine. You can also control the remote computer by using your mouse and such. You see the entire remote "desktop" (assuming your using a GUI) and you can interact with it as if your screen and mouse were plugged into the remote machine. VNC is usually pretty performant when compared to X forwarding.
The primary differences:
- X only allows you to forward "programs" and not the entire desktop
- VNC only allows the entire desktop, which can be annoying if you just need a tiny window
- In general(I think) with X forwarded programs, you can only run them in one place or another. So, you'd have to run two instances of the program if you wanted to see the display on both your local machine and the remote machine.
The suggestions below to export the X output are sound, except they will take up a lot of bandwidth, and may not be ideal, even over the LAN. A better solution may be to set up a video streaming server so that it streams compressed data - Google "linux live streaming video usb" for some solutions for this. [ I know this is not exactly what you asked, hence its a comment rather then an answer ] – davidgo – 2015-10-22T08:23:07.170