When I rent a Linux virtual private server I can connect via the command line and control the computer remotely. What is the equivalent of this called so I can control a Windows computer GUI remotely from a Linux command line? I have tried googling a tons of terms and phrases but it's hard to find any information for something you don't know the name of - or that might not even exist.
If you just want to connect via SSH you need to install a ssh daemon for windows . here you can find a list of SSH daemons.I would suggest you OpenSSH for Windows.But this way you can only run shell commands in the remote server.i.e,You may not be able to interact with GUI this way since there is no X windows forwarding in windows.
You're asking about the wrong thing. The remote access is not specific to VPSes in any way.
On Linux, the program that allows connecting remotely is almost always SSH ("Secure Shell"), with OpenSSH being the most common implementation (Dropbear, LSH, and SSH.COM are a few others). Some systems might have Telnet or even rlogin, but those are almost universally extinct due to absolute lack of security in the modern world.
It's somewhat more complex on Windows, as not all administration tasks can be done from command line (even though this has improved very much in the past few years). The two built-in remote-access features are Remote Desktop (aka RDP or "Terminal Services") for graphical desktop access, and PowerShell Remoting for command-line (specifically PowerShell commands).
You can also find third-party SSH servers for Windows, such as Bitvise WinSSHd, or FreeSSHd, or Cygwin OpenSSH. Older Windows NT versions didn't have PowerShell, but had a built-in Telnet server.
And to answer the question literally, the Windows equivalent to a Linux VPS is… a Windows VPS.