If a hypothesis is falsifiable, then a person can either disprove it entirely, or go through all the possible avenues to disprove it and fail to disprove it. Many scientific truths, such as Einstein's Theory of Relativity, are not, in fact, proven. This is because many people have tried to disprove it, but none have yet succeeded. A simple example of a falsifiable hypothesis is the statement that "John has six eggs in his refrigerator." A search of the fridge can rule out all possibilities of disproving this statement, and so the hypothesis is falsifiable.
A non-falsifiable hypothesis cannot be disproved. This is not because it is inherently accurate, but because a person has no avenues on which to go to try to disprove it. Certain fields of study, such as homeopathy, can use hypotheses -- such as water molecules remembering the imprint of a medicinal substance before the substance undergoes many dilutions, and having a biological effect on the body thereafter. Since possible avenues to disprove this do not currently exist, the hypothesis is a non-falsifiable one.
Falsifiable hypotheses are extremely useful in the field of science, compared to non-falsifiable hypotheses. Falsifiability is a measure of accuracy, and Karl Popper used this characteristic of the informed guess method to differentiate the hard sciences, such as math and physics, from other science fields, such as psychology or cultural anthropology. Only a falsifiable hypothesis can add definite knowledge to the hard sciences, although many people who work in other scientific fields use non-falsifiable hypotheses as part of a less precise understanding of how complex systems in the field, such as trauma in psychology, function.
The Raven Paradox is a thought experiment that shows the line between a falsifiable and a non-falsifiable hypothesis is sometimes blurry. A statement that all ravens are black, is a hypothesis. It is non-falsifiable in the way that a person can go out and check lots of ravens to ensure that they are all black, but never be able to check the entire global population of ravens. It is also falsifiable because there likely is at least one single albino raven in the entire world, and ravens can have albino chicks, disproving the hypothesis. In practicality, however, all ravens are black, so the hypothesis can stand as a useful statement.