There is a program called VNC that is free and will do exactly what you want. It was originally developed by AT&T.
https://www.realvnc.com/ The software runs as a daemon on the machine that you want to control. The daemon must be running in the background in order to connect. You can configure it to run on startup. On the machine that you will work at, you run the "VNC Viewer" application and type in the machine name or IP address of the machine you wish to control.
This solution works great for networks behind a firewall. You don not want the VNC service to be facing an internet connection though. It is not considered secure. Also, VNC will not work through a firewall unless you specifically forward the port from the firewall to the machine to be controlled. Again, this is not recommended.
If you need to control the machine from a remote location, you can use the free service from logmein.com. It works through firewalls and you can add as many computers as you'd like to the account. I use this for a lot of my clients.
Feel free to hit me up with any questions. I've used these systems a lot.