Making a contact within an organisation a Primary Contact means that when ever you send an email, for example to an organisation, it will be sent to the primary contact within that organisation.
To make someone a Primary Contact, go to Contacts> Organisations. From there, select the organisation of which the contact is part of and there will be a list of contacts linked to that particular organisation towards the bottom of the page. Simply click Make Primary.
Or, if no contacts are yet linked to the organisation, link at least one and make it the Primary Contact.