Is "empty set" an element of a set?

The empty set can be an element of a set, but will not necessarily always be an element of a set.

E.g.

$\emptyset \in \{\emptyset, \{a\},\{b\},\{a,b\}\}$

$\emptyset \in \{\emptyset, 1, 2\}$

$\emptyset \in \{A\}$ when $A=\emptyset$

There exist many sets though which the empty set is not a part of:

$\emptyset\not\in \{1,2,3\}$

$\emptyset\not\in \{x,y\}$

$\emptyset\not\in \emptyset$

$\emptyset\not\in\{\{\emptyset\}\}$

What will be true however is that the empty set is always a subset of (different than being an element of) any other set.

$\emptyset \subseteq \{1,2,3\}$

$\emptyset\subseteq \{a,b\}$

Additional details spawned from conversation in comments.

$\emptyset$ is the unique set with zero elements. $\{\emptyset\}$ is a set with one element in it, the element namely being the emptyset. Since $\{\emptyset\}$ has an element in it, it is not empty. $\emptyset\neq \{\emptyset\}$

A set $A$ is a subset of another set $B$, written $A\subseteq B$, if and only if for every $a\in A$ you must also have $a\in B$. In other words, there is nothing in the first set that is not also in the second set.

Here, we have $\{\emptyset\}\not\subseteq \{1,2,3\}$ since there is an element of the set on the left, namely $\emptyset$, which is not an element of the set on the right.

The answer is "It depends". Some sets have the empty set as a member, other sets (like the example) you have given it isn't a member.