rel-author
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rel-author is a microformat for linking a page to a URL representing an author of that page, a brief (similar to OGP) way to indicate authorship.
For the broader discussion, examples, how tos etc. for indicating and discovering the author of a page, see:
How to
You can use rel=author on both <link> and <a> elements on pages written by an author (e.g. post permalinks, feeds like h-feed).
Simple link in <head> like OGP meta tags, to an author’s site:
<link rel="author" href="https://tantek.com/" />
If your site also supports WebSub, you can use the same <link> element for the self relation as well:
<link rel="author self" href="https://tantek.com/" />
Or for a link to an author’s profile on a multi-author site:
<link rel="author" href="https://fedi.example.com/@username" />
Questions
How can I verify the author
See: authorship-spec#Spoofing.
Can you combine rel author with h-card
Q: What is the meaning of the combination of rel="author" with class="p-author h-card" as a one-line authorship inside an h-entry?
Example:
<a href="/" class="p-author h-card author-icon" rel="author" title="Tantek Çelik"><img src="../logo.jpg" alt="Tantek Çelik"></a>
It's not clear to me why rel="author" is used in this instance, but not in other, more verbose, h-card instances. Is the implication of the combination, "The h-card is found at this URL" instead of "This is the h-card"? So far I have interpreted the class="p-author h-card" to mean "This is the h-card." 
gRegor Morrill
A: rel="author" is for legacy parsers that only support rel-author + rel-me for determining authorship, and don't support h-card yet (e.g. Google spiders). It is easy enough to add the rel-author to one-line h-cards.
In more verbose h-cards, using rel-author would add complexity and the cost of that complexity is likely worse than ignoring legacy parsers. It's a trade-off, but the recommendation is to not use rel-author with more complex h-cards.
Answer summarized from chat logs