Remote desktop is a software system which allows you to remotely access a computer's interface from another computer. The interface of the remote system appears on the local computer as a window, like any other application.
Advantages:-
you can copy and paste between the two computers
the other computer don't need to be in the same room
the remote desktop computer doesn't need a graphics card
you can remote into the remote desktop from any computer running compatible software, even a laptop
Disadvantages:-
you need to be running a specific version of Windows (unless you're running 3rd party remote desktop software)
the two computer needs to be at least on the same network
you need to power up both computers if you need to access the one running the remote desktop server
any graphics card installed in the remote desktop machine is rendered useless, during remote desktop sessions, as it isn't used
A kvm switch is a piece of hardware to which you connect your keyboard, mouse, and monitor and then to your computers. Although it has no software requirements it does require that your systems have a matching set of keyboard, mouse and video ports so the switch can be connected to both computers. The older switches supported only vga and ps2 connections, the newer switches support dvi, usb and analogue audio for speakers and microphone or a headset.
Advantages:-
As the name suggests, it is a switch, there are no software requirements for it.
It directly switches your keyboard, mouse and monitor between devices so they are exclusive to the device to which the switch is set.
You don't need to power up both computers if you need to just access one of them.
You get full use of all the installed hardware of both computers.
Disadvantages:-
It's harder to copy between the computers as the clipboard (cut, copy, paste) cannot be used
The two computers need to be located local to each other and to you.