What is an author for the following two critics: Facoult and Barthes? What are the differences and connections?
Let me start, of course, but recommending that you read the two short essays by the critics you name, Barthes' "The Death of the Author" and Foucault's "What is an Author?". But you want answers, of course, and not simply recommendations, so I'll write a little more!
The link provided below gives a pretty substantial discussion of these two works and some points of simiilarity and difference.
For me, the greatest similiarity is that both critics are challenging the dominant concept of the "author." Even today, after decades of such challenges, many people continue to believe that the author is everything -- e.g. the author is in complete control of the text, the reader must know something about the author in order to understand the text, the "author's purpose" (a recurring item on standardized tests in schools in the United States) is something that deserves our full attention, and so on.
Again, for me, the greatest difference is what the two critics emphasize in their pieces. Barthes wishes to shift everything from the author to the text and the reader, whereas Foucault wishes to explore the various historical "functions" of the author. This difference, to me, says everything about the differences between these two critics. Barthes (in his poststructuralist years, at least) is all about the "pleasure of the text," for example, and Foucault is all about how knowledge and experience are organized (e.g. his famous study The History of Sexuality).