Hey MissSweeti,
Firstly, can you try "/savemap test" in game - this will attempt to save a map named test.cgz. The purpose of this is to see if there are actually permissions to write map files. If this does not work try what I have written next.
Did you install Assault Cube in the right place? When you download the game the installer should ask you where to create the appropriate game folders - perhaps you mistakenly clicked on the first option "Windows XP" rather than the second one, which uses "My Documents". You should reinstall Assault Cube.
Firstly, can you try "/savemap test" in game - this will attempt to save a map named test.cgz. The purpose of this is to see if there are actually permissions to write map files. If this does not work try what I have written next.
Did you install Assault Cube in the right place? When you download the game the installer should ask you where to create the appropriate game folders - perhaps you mistakenly clicked on the first option "Windows XP" rather than the second one, which uses "My Documents". You should reinstall Assault Cube.