Somehow you've had a script or something similar deliver mail to your account on your local machine which is stored in /var/mail/$USER
You can either remove the file /var/mail/$USER which will remove the alert that you have mail (as it will alert you if that folder is present and/or has unread messages) or you can use mail to read the messages before deleting them.
Where is this mail?
It's likely to be in the spool file: /var/mail/$USER or /var/spool/mail/$USER are the most common locations on Linux and BSD.
(Other locations are possible – check if $MAIL is set – but by default, the system only informs you about /var(/spool)/mail.)
Usually the spool file is in a very simple mbox format, so you can open it in a text editor or pager.
For a slightly more convenient way, most distributions come with a program called mail (or Mail, mailx). You can try mutt or alpine; you can even configure it to be sent to an outside mailbox. (See "is this real mail?" below.)
What does it contain, and who/what sent it?
Most often the messages contain output of cron jobs, or a system security report by logwatch, or similar junk. Read it and find out.
How important is it?
Depends greatly on the contents of each message.
You should at least scan the subject headers – often people ignore the mail for months never realizing that their daily cron jobs fail.
Is this even actual "mail" in the same sense as email? Or is it just my system telling me something?
Yes to both – it's generated by your system telling you something, but it's also actual email and can be handled as such.
You can (and should) configure your mail software – the "MTA" aka /usr/sbin/sendmail – to forward the messages to your personal mail address. The exact instructions vary depending on which MTA (if any) you have installed, whether this is a personal computer or a server, whether you have your own domain or use a @gmail.com, and so on.
Note that /usr/sbin/sendmail nowadays is a shared API and doesn't necessarily mean the original Sendmail MTA. In fact, you shouldn't use Sendmail, but something more modern like OpenSMTPD, Postfix, or Exim4. All of them provide the same /usr/sbin/sendmail tool, but they're easier to configure, more secure, and just as powerful.
Best Answer
The old school way is to open a terminal and use
at the prompt, then
to delete one message or
for for four messages.
Then
to quit.
If you don't want to delete them unread, just type Enter at the mail prompt and it'll show each message in turn.
Use
for all the details.
Of course, other more sophisticated mail clients may be available, but that will do it.