Quantum cryptography relies on elaborate physical machinery to execute cryptographic protocols whose security rests upon axioms of quantum mechanics (theoretically, anyways).
To quote the wikipedia entry on the BB84 protocol:
The security of the protocol comes from encoding the information in non-orthogonal states. Quantum indeterminacy means that these states cannot in general be measured without disturbing the original state (see No cloning theorem).
There is a good question and answers about "What makes Quantum Cryptography secure?" on crypto.stackexchange. They are verbose, so I will refrain from copying the content here.
Differences between Quantum Cryptography and Modern Cryptography
Quantum cryptography requires specialized machinery in order to execute a run of the protocol. This is a non-negligible disadvantage compared to modern cryptography. If you want to use Quantum Cryptography, you'll need to pay one of the commercial entities that offers the service.
Modern cryptography uses mathematical algorithms implemented in software, which can be performed by any old computer with sufficient resources (which are almost all computers in this day and age). The outputs of the algorithms can be transmitted via an arbitrary communications medium.
If you see a green padlock next to the URL in your web browser, it means your connection to this very site is being secured by modern cryptography - which is effectively being done for free, as far as you were concerned.
Quantum cryptography is often thought to be unconditionally unbreakable due to the laws of the universe. This sounds too good to be true, and it unfortunately is. There is nothing to stop someone from waiting for you to receive your message, then to threaten you until you reveal what the message was. There is also the issue of an adversaries ability to tamper with the hardware. For a rather scathing but in-depth review of these points, see the blog post at cr.yp.to.
Basically, as with all provably secure cryptographic techniques, these guarantees are only provided within the framework of assumptions that the proofs rest upon. An adversary who finds a hole in these assumptions can circumvent the theoretical guarantees that the algorithms offer. That's not to say that QC is totally worthless and overtly non-functional, but that "provable security", as always, needs to be understood to rest on certain sets of assumptions that could be violated in practice.