The lemma states that if an adversary (typically a probabilistic Turing machine), on inputs drawn from some distribution, produces an output that has some property with non-negligible probability, then with non-negligible probability, if the adversary is re-run on new inputs but with the same random tape, its second output will also have the property.
This concept was first used by David Pointcheval and Jacques Stern in "Security proofs for signature schemes," published in the proceedings of Eurocrypt 1996.
[1][2] In their paper, the forking lemma is specified in terms of an adversary that attacks a digital signature scheme instantiated in the random oracle model.
[4] The forking lemma has been used and further generalized to prove the security of a variety of digital signature schemes and other random-oracle based cryptographic constructions.
(Note that since both hJ and h'J are chosen randomly from H, then if h is large, as is usually the case, the probability of the two values not being distinct is extremely small.)
The forking lemma is of use when it would be possible, given two different random signatures of the same message, to solve some underlying hard problem.
Schnorr also suggested enhancements for securing blind signatures schemes based on discrete logarithm problem.