Classical computers work with bits of information. At the most atomic level there is something that is true or false, 1 or 0, black or white.
Quantum computers on the other hand take advantage of phenomena in quantum physics and work with qubits. There are 0 and 1 and everything in between.
They allow applications where instead of doing a calculation for for a large number of possibilities one after the other the calculation is done for all possible inputs all at once.
This is a great explainer from 2016
The term quantum supremacy means a point in time where quantum computers have reached a point that they can do a calculation significantly faster than even the biggest baddest super computer could.
So far Google and the University of Science and Technology of China have claimed to reach quantum supremacy.
Google: Demonstrating Quantum Supremacy
so what is this super fast computing good for?
quantum computers will be capable to break most of today's encryption
simulation of quantum systems in particular for material science , drug discovery, genomics
in other words, it will allow new levels of achievements in several emerging technologies:
3D Printing AKA additive manufacturing
Will it replace "normal" computers?
Don't expect everyone to replace their PC or Mac with quantum computers. Classical 0-1, true or false binary computers will co-exist with quantum computing. Each has its own use and benefits.
The same way how early mainframes were connected with computer networks that grew to become the Internet there is ongoing work and research to transmit information in the form of qubits connecting quantum computers.
One application is secure communications that is physically guaranteed to remain private.
Get Hands On
Cloud providers now each allow people to access quantum computers
Quantum Computing Service—Amazon Braket–Amazon Web Services