A quantum computer is a computer that uses the properties of "individual" fundamental particles and the physics that govern them, known as quantum mechanics, to form a qubit as its basic computing unit. This is in contrast to classical von-Neuman computers that use a "large" number of electrons to build logic circuits. Classical computers are also governed by the physics of quantum mechanics, but classical computers, as far as we know, can not exhibit superposition, which is at the heart of quantum computers. We use the properties of ensemble of electrons to build gates that are then assembled into classical computers. The standard classical digital computers uses the bit.
The idea behind a quantum computer is somewhat complicated, but essentially it uses the weirdness of superposition of quantum particles to explore solutions in a massively parallel way. Classical computers are essentially finding solutions serially. Here is a reasonable explanation:
http://www.dwavesys.com/quantum-computing
and
https://uwaterloo.ca/institute-for-quantum-computing/quantum-computing-101
It is important to understand that a computing unit can be built from just about anything. The reason we use electronic digital computers is that they are fast, and quantum computers are much much faster for a certain class of important problems.
Quantum Cryptography also have another perhaps even more important application, guaranteed secure communications. You can send a message, and know whether it is being tapped in between. Here is a readable article on quantum computers and quantum cryptography one after the other in a concise way:
https://blog.kaspersky.com/quantum-computers-and-the-end-of-security/
That sounds like a great idea except for the fact that every single computer is already a quantum computer and was built from the theory of quantum mechanics.
Now quantum cryptography is a whole another topic of discussion. Of course I don't know you would want to try to achieve this, since it has already been done.
You know, everybody always talks about Einstein and how great the theory of relativity is, but the truly great scientific breakthrough of the 20th century was the theory of quantum mechanics. The real technological break throughs were the result of quantum mechanics, and not so much Einstein.