A pseudo-random function (PRF) is a family of deterministic functions indexed by a parameter, such that a randomly selected instance is computationally indistinguishable from a uniformly random function with the same input and output spaces.
A pseudorandom function family is a finite set of deterministic functions that share the same given (finite) input and output spaces, indexed by a parameter which selects the exact function. Pseudorandomness is achieved if an instance of the family, obtained with a uniformly random selection of the index parameter, is computationally indistinguishable from a function selected at random and uniformly among the whole set of possible deterministic functions with the same input and output spaces.
PRF are most useful when they are efficiently computable. There is no theoretical guarantee that PRF can really exist, but many candidates are known, which cannot be distinguished from random functions with non-negligible probability by attackers with finite computing power. A common example is HMAC/SHA-256; the key is then the selection parameter, with a size large enough to thwart exhaustive search.
