What is the difference between First party and third party GA?
GA primarily utilises 1st party JS cookies. If you have GA on example.com then the cookie will be stored against example.com. One of these cookies stores a unique ID for you as a user.
GA will then send 'beacons' (GET requests to Google endpoint, with data passed in URL parameters) for pageviews and other events. Your unique ID is included in these URL parameters, which allows the various events from you to be joined up into a 'session'. That has not changed.
Apple's ITP affects both 3rd party HTTP cookies and 1st party JS cookies.
The main impact of ITP on GA is to limit the age of these cookies to auto delete after 7 days (usually), meaning that if you leave a site for more than 7 days your new session won't be able to be joined up with your previous session.
There is an edge case where ITP 3rd party cookie limits will directly block GA JS cookies, which is when it is running inside an iFrame (hat tip to Simo [1] for alerting me to this), in which case the JS code may not be able to set a cookie and will not fire a beacon.
I believe that this would limit GA's ability to store data in cookies on my system (user or session-based attributes). GA could still create unique identifiers based on my user or session-based attributes and store those IDs on their own servers, then call them back up when I revisit the site.
I'm still coming up to speed on this, maybe someone with more experience could correct or clarify my statements.
Google Analytics stores a first-party cookie with a unique ID for each user. This is used to de-duplicate traffic from the same user to the same domain. This cookie not shared across websites. If you go to example.com and website.com, those are two different first-party cookies without views into each other.
Optionally, Google Analytics can be configured correlated this with a third-party cookie from google.com for advertising purposes. This allows the website owner for example.com to connect a user converting on example.com to that same user having seen or clicked an ad on google.com. It also allows the website owner to target ads on google.com towards users based on their activity on example.com
Previous iterations of Safari's ITP have outright blocked the third-party cookie, and have limited first-party cookies based on whether they're HTTP cookies or JavaScript cookies.