Sets contain elements. There is no rule that says these elements can't be sets themselves.

The power set P(A), is defined as a set whose elements are all the possible subsets of set A. Hence, each and every element of P(A) is a set itself.

When elements are not sets, we use lower case letters to denote them. When they are sets, then we use upper case letters to describe them.

So X is just a set, which is a member of the set P(A). And hence the elements of X are elements of A. X is a subset of A.

So you are being asked to find the set

This translates, in english as:

Find S, which is the set of all sets X, which are members of the power set of A, such that the cardinality of X is equal to the cardinality of the complement of X.