Blender's windowing system

Blender's "innovative non-overlapping window system" is probably a bit different than the windowing systems that you're used to.  As stated, Blender windows do not overlap each other.  The default window layout in Blender looks something like this:

Each Blender window has an accompanying "window header" containing options related to that window.  Window headers may be located at either the top of the bottom of each window.

Moving window headers

A window's header may be moved to the top or bottom of the window by right clicking on the header and choosing the appropriate option from the "Header" menu that appears:

Window headers may also be turned off using the same menu.  Note: A window's header may be re-enabled by left-clicking on the dividing line between windows and choosing "Add Header" from the menu that appears.  The mouse cursor should be slightly more towards the window whose header you wish to re-enable (if the double arrow cursor goes away, the mouse cursor is too far towards the window).

Changing the type of a Blender window

The window type selector on the left-hand side of a window header may be used to change the type of any Blender window

Resizing Blender windows

Blender windows may be resized by moving the mouse window over the line that divides two windows.  When the mouse cursor is on this like, the cursor changes to a double arrow:

When the cursor changes to a double arrow, hold down on the left mouse button and drag the dividing line between the windows.  Release the left mouse button to complete the resize operation.

Splitting Blender windows

Blender windows may be split into two separate windows by moving the mouse cursor over any edge of a window and right-clicking when the "double arrow" cursor appears:

After choosing "Split Area" from the menu that appears, move the mouse cursor to the desired location and left click to complete the operation.

Maximize and restore a Blender window

Any window may be maximized or "Full screened" by moving the mouse cursor over the desired window and pressing "ctrl-uparrow" on your keyboard.  The window may be restored to its original position (un-fullscreened) by pressing "ctrl-downarrow".

Using the predefined window layouts and defining new layouts

Blender comes with several predefined window layouts that may be selected using the "Layout" pulldown in the info window header at the top of the Blender window:

You can cycle between the layouts using the "ctrl-leftarrow" and "ctrl-rightarrow" keyboard shortcuts.  You may customize these layouts and/or add your own.

Custom window layouts may be saved so that Blender remembers them when you create a new .blend file.  When you are happy with your custom layout(s), click "File->Save Default Settings" in the info window header at the top of the Blender window.  Note: It's best to be sure that the scene is empty or only contains the default cube, lamp, and camera objects before saving new default settings.  The contents of the current Blender scene are also saved as the defaults, meaning that any objects in the scene will also show up when creating a new .blend file.