Are all functions sets? I am studying Zermelo-Frankel set theory from Jech's Set Theory book. I understood it like functions are sets but the book uses the phrase "If a class F is a function" in Axiom Schema of Replacement. Why does it call it a class F instead of a set?

You're right that technically that's a bad phrasing. It's informal language to help motivate the axiom, which is a bit technical.
More precisely, Replacement says:

If $F$ is a class such that $(i)$ each element of $F$ is an ordered pair, and $(ii)$ for each $(x, y), (x, y')\in F$ we have $y=y'$, then [the rest of the axiom].

A class satisfying $(i)$ and $(ii)$ is called a class function; basically, Replacement says "any class function whose domain is a set, is a set" (actually it says that the range is a set, but it's easy to see that these statements are equivalent).
