This definition does not involve nondeterminism and is analogous to the verifier definition of NP. See FP for an explanation of the distinction between FP and FNP. There is an NP language directly corresponding to every FNP relation, sometimes called the decision problem induced by or corresponding to said FNP relation. It is the language formed by taking all the x for which P(x,y) holds given some y; however, there may be more than one FNP relation for a particular decision problem.
Many problems in NP, including many NP-complete problems, ask whether a particular object exists, such as a satisfying assignment, a graph coloring, or a clique of a certain size. The FNP versions of these problems ask not only if it exists but what its value is if it does. This means that the FNP version of every NP-complete problem is NP-hard. Bellare and Goldwasser showed in 1991 using some standard assumptions that there exist NP-complete problems such that their FNP versions are not self-reducible, implying that they are harder than their corresponding decision problem.
FP = FNP if and only if P = NP.