A 320x240x256 resolution, unlike 320x200, provides square pixels. With pixels this big, it's important.

Unfortunately it requires more than 64k of ram, so mode X (as this was sometimes called) required the pixels to be separated into bit planes. (In DOS/real mode there was only a 64k block of memory addresses alotted to the VGA card.) A single pixel is arranged like this: (shown compared to 320x200)

       Mode X                      Mode 13
 /0/4/____   plane 0
^^^^^^^^^^                      (only one plane)
 /1/5/____   plane 1           /_______________
^^^^^^^^^^                    /0/1/2/3/4/5/6/7/___
  /___                       ^^^^^^^^^^^^^^^^^^^^^
 /2/6/____   plane 2
 /3/7/____   plane 3
where only one plane is paged in at a time. Each byte holds 2 bits of 4 pixels. This way, only 19k of contiguous memory is required.

But yeesh... that's pretty messy. Is it any wonder no one uses bit planes anymore?

On newer, VESA compliant video cards, more than 64k is allowed through windowing in real mode, and in protected mode there is no 64k limit.