Is there any app or hack which can split a monitor into multiple screens? So for instance, a 1920x1080 display could be treated by the operating system as though there were two 960x1080 screens connected instead.
There is No: True Multi-Monitor Support for a Single Display
There is no way to get the OS to treat a single display as more than one display.
You Can: Fill Defined Sections of your Screen with an App
Adding my vote for Moom! Excellent - as simple or complex as you need it to be, and keeps out of the way. http://manytricks.com/moom/
This is a tool that, like the ones mentioned in the other answer, lets you resize windows to fit defined sections of your screen.
Since writing this answer, I have switched to using Magnet which is even simpler and more modern; matching my needs. https://magnet.crowdcafe.com
Since El Capitan, you can zoom two windows to "full screen" in the same Space: https://support.apple.com/en-us/HT204948
To the best of my knowledge, you can't rearrange the two applications within the Space (for example, splitting vertically instead of horizontally) other than swapping them side-to-side, nor have more than two windows in the same Space.