Unsigned code refers to an application which has not been signed with the secret key required for it to load on a console.
In the video game console business, most console games have to be signed with a secret key designed by the console maker or the game will not load on the console. There are several methods to get unsigned code to execute which include software exploits, the use of a modchip, a technique known as the swap trick or running a softmod.
It may not initially seem obvious why simply copying a signed application onto another DVD does not allow it to boot. On the Xbox, the reason for this is that the Xbox executable file (XBE) contains a media-type flag, which specifies the type of media that the XBE is bootable from. On nearly all Xbox software, this is set such that the executable will only boot from factory produced discs so simply copying the executable to burnable media is enough to stop the execution of the software.
However, since the executable is signed, simply changing the value of the flag is not possible as this alters the signature of the executable causing it to fail validation when checked.