A Linked Contact is a contact who appears in multiple follow-up periods. If you have migrated an individual contact, you will see a note on their personal page reminding you of the periods where they exist prior to the active one. The note will look like like this:
Creating linked contacts allows you to keep the same contact information, but track follow-up data in a new period all together. Essentially you give the contact a clean slate with regards to events attended, follow-up status, etc. When you migrate a contact and they begin a new period, their data will be connected to analytics for the active period, not the old period.