Let me unravel these two one by one.
Page Authority is Moz's calculated metric for how well a given webpage is likely to rank in Google.com's search results. It has a tight relation with domain authority, which measures the strength of entire domains or subdomains.
Any site can be popular, but what matters is if people link to it. For instance, a porn site is popular, but not many people link to it. As for a government website, it has authority, because people tend to link to it.