Quantum cryptography is a scientific discipline that exploits the laws of quantum mechanics to perform cryptographic tasks, e.g., secure message transmission. It turns out that quantum phenomena are in fact very useful for that purpose: The level of security that can be achieved by quantum cryptographic devices is usually much higher than it is for purely classical schemes.
Maybe the most popular example of a cryptographic task is the transmission of a secret message over an authentic channel: Consider two parties, traditionally called Alice and Bob, who are connected by an authentic, but otherwise insecure, classical communication channel. Authentic means that it is impossible for an adversary, Eve, to modify the communication between Alice and Bob. Assume now that Alice wants to send a secret message to Bob, i.e., Eve, who might be wiretapping the communication, should be unable to get any information on this message.
In classical cryptography, this can be achieved with a public key encryption scheme such as RSA. However, public key encryption merely provides computational security. This means that it is computationally hard, but not impossible, for an adversary to get information on the secret message. It seems that this is actually the best one can hope for. In fact, it follows from an argument of Shannon and Maurer that information-theoretically secret communication is impossible without any further assumptions. If one additionally assumes that Alice and Bob, in addition to the authentic classical channel, have at hand some (completely insecure) quantum channel, the situation looks quite different. In fact, using a quantum key distribution scheme, Alice and Bob can agree on an arbitrarily long secret key. This key can then be used for one-time pad encryption which is perfectly secure.
Inspired by this sucess, researchers made a considerable effort to realize other cryptographic tasks such as bit commitment. Unfortunately, it turned out that quantum mechanics is not as helpful in that case. However, there are some proposals for quantum protocols which have considerable advantages compared to any classical scheme.
In contrast to quantum computers, quantum cyrpto devices are realizable with today's technology and are even available commercially.