Sorry for the late answer: I had a busy week.
Short answer: Various ways.
Long answer:
Windows: RDP, Remote Desktop Protocol. Included in every Windows Version that is Pro or better (Education works too). Non-Pro Versions can only access, not be accessed.
VNC: A free protocol for using a desktop session over the network. There are a lot of working implementations (vnc servers and vnc clients) I think I usually use thightvnc under windows. Just get the GNU Version and you should be golden (other, non-free implementations might have some restrictions).
Commandline Access:
Windows - to - Windows: Powershell.
Just kidding unless you plan on being a Windows Admin there are more fun ways to spend your day.
Linux to Linux:
SSH. Plain and simple. Most mainstream distros ship with a ssh client installed by default. A ssh server is sometimes installed and enabled.
Windows to Linux:
Any SSH client will work. Commonly used is Putty but I end up using cygwin because it works like linux (i.e. I can use the same configuration file as in my linux machines). The Linux subsystem in Windows works too*.
*establishing a connection takes a couple of seconds sometimes in my setup, no clue why that is, your mileage might vary.
As for the wiring:
Some solutions in order of connection latency/bandwith
a) Ethernet cable: Something that supports 1 Gigabit now and maybe 10 in the future.
b) Wifi either 5 or 2,4 Ghz with little interference by neighbours
c) Powerline Ethernet (or if you are German D-LAN)
d) Crappy Wifi with tons of other wifis in the area
Bonus solution: If you have cheap LTE you could possibly connect over LTE to the computer in the other room if the computer has internet.
Bandwith needs of applications:
RDP / VNC: I have recently metered a connection to determine wether it is viable to use it via my phones hotspot. A Windows RDP session needed several gigabytes an hour even with some tricks (reducing color depth, not scrolling but using the page up and down buttons) but that did not help.
SSH: Next to nothing. Unless you transfer files (then it needs the size of the files in bandwith).
What is your situation in terms of home office or whereever you have space to put the pc?