First-party relationships would be created if the user “intends to interact” with the web company (or a service provider acting on behalf of that company). For example, logging into Facebook would count as a “user action” that would allow Facebook to track your activity “across multiple distinct contexts,” including other websites.
In contrast, companies with third-party relationships would have far more limited tracking abilities. For example, if a user visits a site that integrates an advertisement with content from other sources, the ad server would not be able to place a tracking “cookie” for marketing purposes on your device without your consent.