Over the years I've used a number remote display products.
The first one I recall using was Virtual Network Computing aka VNC in 1998. VNC is free software and has been incorporated, forked and enhanced by a number of companies and projects. While VNC works rather well over a medium to high speed LAN, it can be a bit slow over slower connections. That is because VNC isn't a completely vector-based protocol and still uses mostly bitmaps... or at least that is my understanding. One good thing about VNC though is that both the client and the server are available for Windows, Mac, and Linux... and a few other things as well.
Let's examine a few alternatives.