It depends on what you want to do exactly with this setup.
Assuming I understood what you meant, one solution could simply be to hook both monitors to the computer (you'll need a suitable video card), setup two desktops, and run a virtual machine using VirtualBox (free) or VMWare or a similar application on the second desktop. The base layer (that is, the operating system running the virtual machine) would be "Computer 1", and the virtual machine would be "Computer 2".
You will go back and forth between machines by moving the mouse. Depending on the VM and the operating system installed therein, the movement will be fluid and seamless or the virtual machine will "capture" the pointer.
Since Computer 1 (base layer) will show up on Monitor 1 and Computer 2 will show up on Monitor 2, they will appear and act, in practice, like two separate computers.
Only, not really. First off, they're sharing the same hardware, so the performance of both will be middling at best. In short, you can probably forget about, say, playing a latest-gen 3D game on "Computer 1" while the virtual machine (that is, Computer 2) is running.
Also, the Virtual Machine would still be dependent on the Main Machine, so if this goes offline, the Virtual Machine would go offline as well, just like the several personalities of somebody suffering from Multiple Personalities Disorder would all disappear should the sufferer die.
More could be said, but you should really expand on your question a bit more, giving us more info about your operating system(s), your hardware, the software you want to run, and so on. Chances are that a simple 1 computer, 2 desktops configuration may suffice.