Basically, this is just a "You've got mail" notification.
Most likely these are system generated logs or errors being mailed to root. When logged into root type "mail" to see them.
Importance wise, it depends on what you perceive as important. I myself like to know what is going on with my system so I leave them on, while others do not.
Disabling them would involve either, disabling postfix all together (or any other mailserver), this would prevent all system mail.
Or, if you are using this as an email server, disabling other things such as logwatch, and any other script which may mail root,
Lastly another option is creating a dumby user for all mail directed at root to go by setting up aliases in /etc/aliases (Basically at end: root: dumbyuser) and then running newaliases.
Hope this helps...