No, you weren’t far off. A single site can only get and set cookies on its domain. For example, joesblog.com can’t read your Facebook session cookie, because that would mean they could just steal your session and impersonate you.
But third-party cookies are when joesblog.com has a Facebook like button on each post. Those resources are hosted by Facebook, and when your browser makes that request, it sends your Facebook cookies to Facebook. But this also lets Facebook know which page you’re visiting when you make that request, which is why people are upset.
With this third-party cookie blocking, when you visit joesblog.com and it tries to load the Facebook like button, either the request or just the request’s cookies will be blocked.
Although that raises an interesting question. Facebook is at facebook.com, but its resources are all hosted under fbcdn.com. Have they just already built their site to handle this? Maybe they just don’t strictly need your facebook.com cookies to load scripts, images, etc. from fbcdn.com.