Falsifiability is the capacity for some proposition, statement, theory or hypothesis to be proven wrong. That capacity is an essential component of the scientific method and hypothesis testing. In a scientific context, falsifiability is sometimes considered synonymous with testability.
In hypothesis testing, the null hypothesis usually states the contrary of the experimental or alternative hypothesis. The null hypothesis provides the basis of falsifiability, describing what the outcome would demonstrate, should the prediction of the hypothesis not be supported by the study. The researcher's hypothesis might predict, for example, that fewer hours working correlates to lower employee productivity. The null hypothesis would be that fewer hours working is correlated with higher productivity, or that there is no change when employees spend less time at work.
The requirement of falsifiability means that conclusions cannot be drawn from simple observation of a particular phenomenon. The black swan problem is an illustration: If a man lives his life seeing only white swans and never knows that there are any non-white swans, he might assume that all swans are white. For falsifiability, it isn't necessary to know that there are black swans but simply to understand that the statement "All swans are white" would be disproven should a single non-white swan exist.
The Austrian philosopher and scientist Karl Popper (1902-1994) introduced the concept of falsifiability in his writings on the demarcation problem, which explored the difficulty of separating science from pseudo-science.
Watch a brief tutorial about falsifiability: