What is user NONE?

None isn't there by default. After clicking Add, entering None and using Check Names, just click Cancel and it should not be added to the list of users, and would have continued being covered by the permissions for the Everyone group (which by default does not include Full Control)... but the Allow boxes should have matched the permissions given to the Everyone group (where it came from) when adding it.  If you delete that specific username, it will go back to being covered by the Everyone group's permissions.

As you're digging, you might also notice there's a Group named None.
e.g. click the Advanced button on that dialog, go into Object Types to uncheck all but Groups, OK back out then Check Names on 'None' (with only "Groups" in the top field) and you'll find that group name too.  :-)

OK, so it won't show anywhere, no matter how hard you try or how many Windows dialogs you inspect, eg. (my standalone PCs Username is DC845WN)

C:\>net localgroup

Aliases for \\DC845WN

-----------------------------------
*Administrators
*Backup Operators
*Guests
*HelpServicesGroup
*Network Configuration Operators
*Power Users
*Remote Desktop Users
*Replicator
*Users
The command completed successfully.

I don't see a Group named NONE, which is why I was puzzled.

Kit Skinner asked the same question here, and it's a damned good question to which I certainly don't fully know the answer and nobody did there either:
http://archives.neohapsis.com/archives/sf/ms/2001-q1/0154.html

Good plain english explanations of SIDs (Security Identifiers):
http://kb.iu.edu/data/aotl.html
http://en.wikipedia.org/wiki/Security_Identifier

More in-depth:
http://servermigrator.blogspot.com/2006/02/why-understanding-sids-is-important.html

Well known SIDs in Windows:
http://support.microsoft.com/kb/243330
Scroll down to this one in the above page:

SID: S-1-5-domain-513
Name: Domain Users
Description: A global group that, by default, includes all user accounts in a domain. When you create a user account in a domain, it is added to this group by default.

Take particular note of that "513" RID (Relative ID - http://en.wikipedia.org/wiki/Relative_ID) at the end of the above SID and then run the SECTOK.EXE utility program by Joe Richards:

SecTok - Command line tool to display parts of the process token of current process. (WinNT to Vista).
http://www.joeware.net/freetools/tools/sectok/index.htm
Displays the user SID and name of the current token and all sids/names of groups that token has.

This utility should display the NONE Group's SID when run from the command line with no parameters, or redirected to a text file:
sectok > report.txt

My results on a standalone XP system - see the "513" RID at the end?:

User: S-1-5-21-527237240-1715567821-1606980848-1003 - DC845WN\Bill

Group: S-1-1-0 - Everyone
Group: S-1-2-0 - LOCAL
Group: S-1-5-11 - NT AUTHORITY\Authenticated Users
Group: S-1-5-21-527237240-1715567821-1606980848-513 - DC845WN\None
Group: S-1-5-32-544 - BUILTIN\Administrators
Group: S-1-5-32-545 - BUILTIN\Users
Group: S-1-5-4 - NT AUTHORITY\INTERACTIVE
Group: S-1-5-5-0-55070 - Logon Session

According to this page (related to Cygwin):
http://www.cygwin.com/cygwin-ug-net/ntsec.html

GID 513 is of special interest. This GID is a well known group with different naming in local systems and domains.  Outside of domains the group is named 'None' (`Kein' in German, `Aucun' in French, etc.), in domains it is named 'Domain Users'.  Unfortunately, the group `None' is never shown in the user admin tool outside of domains.

This is an interesting discussion about Windows Local vs Domain SID Cloning Issues, the lead-in part being the most universally related to this general issue:
http://www.excaliburtech.net/archives/117