Dividing my desk into two work areas

It's called workspaces and if you're running plain old vanilla Ubuntu with Unity, go the to gear icon in the top left of your screen:

then go to:

• System Settings
• Appearance
• the Behaviour tab
• Enable Workspaces

and then press the workspace switcher icon:

and be amazed:

Note: Pressing Super+S¹ accomplishes the same thing.

¹_{Super+S is aka Windows+S}

I agree with the posted answers; they're the right answers. But there's another option in stock Ubuntu (14.04 anyway) that satisfies one interpretation of the question, and can generally be handy to know

You can easily resize windows to take up half or a quarter of the screen. You can do this with Ctrl+Alt+any number on the num pad (in number mode). This will resize the current window as follows:

• ctrl+alt + ...
  • 1: top left
  • 2: bottom half, full-width
  • 3: bottom left
  • 4: left half, full-height
  • 5: full screen
  • 6: right half, full-height
  • 7: top left quarter
  • 8: top half, full-width
  • 9: top right quarter
  • 0: minimize window (I've remapped this to make windows take the full vertical screen height, keeping the current width)

So with one window do Ctrl+Alt+4, and with the other Ctrl+Alt+6, you'll have the screen split into two halves temporarily.

Tilers are excellent choices.

You can install it the following way:

To install awesome on Ubuntu 16.04 (Xenial Xerus) run the following commands:

sudo apt-get install awesome

You can read more about the tilers in this article and on the AWESOME website itself.