Quantum Computing in a nutshell

Recently, I have been approached several times about an easy way to describe what a quantum computer is. Mainly the basic ideas, first explaining the realms of the quantum world, with superposition and entanglement, have to be clear enough to continue with the definition of the qubit.
The biggest challenges in quantum computing, quoting from Igor Markov, EECS Professor at Michigan and currently at Google, can be summarized as follows:
Applied physics

• Long-living qubits for memory and communication
• High fidelity for a universal set of qubit transformations
• Scaling qubit processors to more than a dozen qubits
• All of the above without extremely low temperatures and at a reasonable cost

Theoretical physics and related mathematics (Lie groups, knot theory, etc)

• Quantum information processing with new particles and physical systems
• New quantum models of computations and new types of quantum error-correction
• Effective quantum control in reproducible quantum systems (including the control of quantum decoherence)
• (Algorithms for) simulating interesting physical systems on various quantum computers

Theoretical computer science, algorithms, and related mathematics (number theory, matrix decompositions, etc)

• Coming up with useful challenges that allow for quantum speedup and finding quantum algorithms that achieve a speed-up
• Proving that some problems do not allow for a quantum speed-up
• Evaluating the complexity/power of various quantum models of computation, proof systems, games (sometimes suggested by theoretical physicists), reducing some of them to others

Quantum computer engineering

• Compilation of quantum algorithms info more effective quantum circuits and control sequences
• Simulation and verification of quantum circuits, algorithms and control sequences, etc
• Better quantum error-correction, perhaps tailored to specific quantum algorithms
• All of the above for specific quantum computers developed by applied physicists

Interdisciplinary challenges

• Avoiding excessive hype. Making it very clear that quantum computers are unlikely to run your email or Web browser in the future.
• Getting funding (this may conflict with the above challenges, so some researchers reorder their priorities)

Also, in the picture below you have an interesting recapitulation about quantum computers, and a very nice video explaining the general idea behind and fundamental concepts.
Please do not hesitate to post any question you might have, I will be more than happy to give a, hopefully, helpful answer.