Given a set of integers, FIND-SUBSET-SUM is the problem of finding some nonempty subset of the integers that adds up to zero (or returning the empty set if there is no such subset).
Given a set of integers, SUBSET-SUM is the problem of finding whether there exists a subset summing to zero.
Simply ask the black box to find the subset that sums to zero, then check whether it returned a nonempty set.
In this context NP stands for nondeterministic polynomial time.
There are also NP-equivalence classes of Boolean functions, where NP stands for negation and permutation.