In The Logic of Scientific Discovery, what does Popper mean by the falsifiability of a theory and why is that important?
When Popper says that a theory has "falsifiability," he means that it can be proven false if it is indeed false. He does not mean that the theory actually is false. Even some theories that are (so far as we know) true are falsifiable. A theory is falsifiable if it could possibly be proven false. For example, the theory of gravity could be falsified if, even just once, an object was released in the air, had no forces acting upon it, and yet did not fall toward the center of the Earth.
This is very important to Popper because it defines what is and is not science. Popper argues that only a theory that can be falsified is really a scientific theory. Other kinds of theories cannot properly be classified as scientific because they could be false and yet we would never be able to prove it because they are not falsifiable.