First-party pixels belong to the same website (or domain) you are visiting. If you are in Facebook.com, there will be Facebook pixels gathering data about your activity, location and time spent, among other things.
Third-party pixels (the most common ones) are pixels that belong to tracking tools (like Mixpanel, Google Analytics, and ehem.. Facebook.) that are doing the same monitoring as the first-party, except the users’ activity data is not being collected by the website
There is the promise that while you are being tracked, the website can provide you with a better experience (for example Amazon can recommend your products based on what you have in your cart.)
After GDPR users are becoming more aware of the fact that browsing in the web is like someone filming you with a cellphone while you click, read and jump between pages.
Eventually, same as with cookies, advertisers will have to start asking for permission to do this (i.e., to enable pixels.) and being very clear about what they are going to do with your tapes.