What does the error "X is not in the sudoers file. This incident will be reported." philosophically/logically mean?

The administrator(s) of a system are likely to want to know when a non-privileged user tries but fails to execute commands using sudo. If this happens, it could be a sign of

1. a curious legitimate user just trying things out, or
2. a hacker trying to do "bad things".

Since sudo by itself can not distinguish between these, failed attempts to use sudo are brought to the attention of the admins.

Depending on how sudo is configured on your system, any attempt (successful or not) to use sudo will be logged. Successful attempts are logged for audit purposes (to be able to keep track of who did what when), and failed attempts for security.

On a fairly vanilla Ubuntu setup that I have, this is logged in /var/log/auth.log.

If a user gives the wrong password three times, or if they are not in the sudoers file, an email is sent to root (depending on the configuration of sudo, see below). This is what's meant by "this incident will be reported".

The email will have a prominent subject:

Subject: *** SECURITY information for thehostname ***

The body of the message contains the relevant lines from the logfile, for example

thehostname : Jun 22 07:07:44 : nobody : user NOT in sudoers ; TTY=console ; PWD=/some/path ; USER=root ; COMMAND=/bin/ls

(Here, the user nobody tried to run ls through sudo as root, but failed since they were not in the sudoers file).

No email is sent if (local) mail has not been set up on the system.

All of these things are configurable as well, and that local variations in the default configuration may differ between Unix variants.

Have a look at the mail_no_user setting (and related mail_* settings) in the sudoers manual (my emphasis below):

mail_no_user

If set, mail will be sent to the mailto user if the invoking user is not in the sudoers file. This flag is on by default.

In Debian and its derivatives, the sudo incident reports are logged to /var/log/auth.log which contains system authorization information, including user logins and authentication mechanisms that were used:

$ sudo su
[sudo] password for regularjohn: 
regularjohn is not in the sudoers file.  This incident will be reported.

[as root]

$ tail -n 1 /var/log/auth.log
Jun 21 16:30:26 marvin sudo: regularjohn : user NOT in sudoers ; TTY=pts/19 ; PWD=/home/regularjohn ; USER=root ; COMMAND=/bin/su

This log file is typically only accessible to users in the adm group, i.e. users with access to system monitoring tasks:

$ ls -la /var/log/auth.log
-rw-r----- 1 syslog adm 76189 Jun 21 16:30 /var/log/auth.log

From the Debian Wiki:

Group adm is used for system monitoring tasks. Members of this group can read many log files in /var/log, and can use xconsole. Historically, /var/log was /usr/adm (and later /var/adm), thus the name of the group.

Users in the adm group are usually administrators, and this group permission is intended to allow them to read log files without having to su.

By default sudo uses the Syslog auth facility for logging. sudo's logging behavior can be modified using the logfile or syslog options in /etc/sudoers or /etc/sudoers.d:

• The logfile option sets the path to the sudo log file.
• The syslog option sets the Syslog facility when syslog(3) is being used for logging.

The Syslog auth facility is redirected to /var/log/auth.log in etc/syslog.conf by the presence of the following configuration stanza:

auth,authpriv.*         /var/log/auth.log

Technically, it doesn't mean anything much. Many (if not all) other software logs logins, failed or otherwise. For example sshd and su:

Jun 21 17:52:22 somehost sshd[25807]: pam_unix(sshd:auth): authentication failure; logname= uid=0 euid=0 tty=ssh ruser= rhost=::1  user=root
Jun 21 17:52:22 somehost sshd[25807]: Failed password for root from ::1 port 37268 ssh2
Jun 21 17:52:23 somehost sshd[25807]: Connection closed by ::1 port 37268 [preauth]
Jun 21 17:52:28 somehost su[25809]: pam_unix(su:auth): authentication failure; logname= uid=1000 euid=0 tty=/dev/pts/15 ruser=someuser rhost=  user=root
Jun 21 17:52:28 somehost su[25809]: pam_authenticate: Authentication failure
Jun 21 17:52:28 somehost su[25809]: FAILED su for root by someuser

Also, many systems have some sort of automation for detecting excessive authentication errors to be able to deal with possible brute-force attempts, or just use the information to reconstruct events after problems appear.

sudo doesn't do anything especially exceptional here. All the message means is that the author of sudo appears to have taken a somewhat aggressive philosophy in communicating with users that happen to run commands they cannot use.