Probably the shortest and easiest definition is the following:
in classical computing we use electricity to model bits (0s and 1s) and then perform computations
in quantum computing we use quantum elements (there are different implementations possible: photons, superconducting stuff, etc.) and their quantum mechanical phenomena (entanglement, superposition, etc.) to model something called qubits (complex linear combinations of 0s and 1s) and then perform computations
Basically it is not better or faster or whatever it's simply a different computational model. This is not black magic. lots of buzzwords recently but really this is just another model of computation which is pretty damn interesting to learn and quite mind-bending :)
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