In the book Set Theory of Jech, he defines a class as follows:
And then in the next page, he says,
Every set can be considered a class. If $S$ is a set, consider the formula $x\in S$ and the class $\{x:x\in S\}$.
But in the definition of class,
members of the class $C$ are all those sets $x$ that satisfy $\phi(x,p_1,...,p_n)$.
Does it mean that the elements of a set is always sets?
As Jech says, "ZFC ... has only one type of object, namely sets". The axiom of extensionality says that all objects of the universe behave like sets: for any $x$ and $y$, if $z\in x$ iff $z\in y$ for all $z$, then $x=y$. That is, every object is determined by which other objects are elements of it. In this sense, "everything is a set" in ZFC, and so in the context of ZFC, the term "set" just refers to any object in the universe. There are no objects other than sets which could be elements of sets.