Email notifications on a Linux system are pre-configured to send emails about various system events. On the level of errors is the default setting. Also by default, root@localhost is receiving all of those messages, as it should. However, since you probably don't use the root account very much, it is one the best practices to redirect all root’s email to your own user, or your email account.
As an example, my user's name is Tim, so I would type:
echo "root: tim" >> /etc/aliases
And now root's email notifications will go to my inbox. If it's a server where you don't spend a lot of time on the console, you can do something like the following as well:
echo "root: tim[at]pleasedontspam[.]me" >> /etc/aliases