Is there a way to "split" one display so that it treated as two separate screens? 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
You can get very close to what you are looking for with a tool like SizeUp or Divvy. They will allow you to fill sections of your screen with an app. Halves, quarters, etc.
Update: Answer still largely accurate as of 7/2020
After many years a friend mentioned he came across this post. Here is an updated response:
You still can't do true Linux-like window management or split a large screen into multiple monitors. Mac OS has added to it's Spaces feature including the ability to have fractional spaces (two apps side by side) which is really just a first party version of the third part solution mentioned above. It is still very limited as a window management system and generally less useful than any of the third party tools mentioned in this thread so far.
You can find a pretty thorough walk through of Spaces over at Apple Insider.
Also, today I use Moom (as mentioned by other commenters below). I find it to be the best modern answer to this question.