A linked account is an account that your NetID has been provided access to. For example, after entering your NetID login credentials at the NetID login screen, you will have the option of logging into your primary mailbox or any other mailboxes (accounts) that have been linked to your NetID, such as a departmental account. A NetID can be "linked" to any number of other accounts, simplifying the login process for users and providing much needed affiliation between accounts. Linked access to an account also provides other privileges such as consolidating in preparation for the migration to Office 365.
Instructions for linking an Office 365 service account to your NetID are available here.