Very short answer:
What makes it so cool and amazingly useful is that Binary 1’s and 0’s replaced by two-level system allowing for infinite superpositions of states. Imagine, only 300 qubits is equal 2 to the power of 300 !!. No classical computer can even dream of it!
What can we solve with this thing?
The short answer is everything that classical super computers can't solve. There is a lot of intractable problems we wish to solve; global warming, chemistry, new materials, financial market to name a few.
Longer Answer:
There are two systems. Classical computers like what you are using to read this pot, Quantum computers that don't yet exist at full scale. Classical and quantum computing involves performing operations on elements of a computational basis. With classic computers such as desktop, mobile handset, or even supercomputer farms, integers are usually represented as binary numbers, with each digit encoded in a physical bit or for quantum computer, the qubit. For example, 1 and 0 for the classical, both 0 and 1 (spin up or down) for the quantum computer, but in superposition mode, which can exponentially increase due to superposition of qubits in all states.
The difference between the type of computation available to a quantum computer and a classical machine emerges from multipartite quantum entanglements. The entangled states manipulated by a Quantum computer physically represent correlations between elements of the computational basis.
"Quantum entanglement is a physical phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of the others, including when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical and quantum physics: entanglement is a primary feature of quantum mechanics lacking in classical mechanics."
A Quantum Computer is capable of producing entangled states via processes of following formula:
where, as mentioned above, the superposition condition on the right-hand side of formula fully represents the correlations between the integer x and the output of the function f(x) - a measurement of this state will always produce a pair of values correlated such that if one is |xi the other is |f(x)i, for any x in the superposition state) without fully representing the 2 N − 1 values which constitute the complete (x, f(x)) truth table (since there is only a small probability that a measurement will extract any given pair of (x, f(x)) from the state).
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