Falsifiability is a fundamental property of a statement (or hypothesis, conjecture, or theory) that it is possible to be demonstrated to be false. According to the very influential philosopher of science Sir Karl Popper, a claim cannot be "scientific" (or subject to scientific inquiry) unless it is falsifiable. Although in the opinion of some, Popper's definition is incomplete, it remains undeniable that falsifiability is an important part of science: a hypothesis that cannot be falsified is a hypothesis that makes no difference in the universe.
In the philosophical community, the term "falsify" has come to mean "disprove", opposite "verify". It's unfortunate that this term has become so entrenched, as outside philosophy the term means "to counterfeit" - to dishonestly make a false representation of something.
"Falsifiable" does not mean "false." For a proposition to be falsifiable, it must be possible, at least in theory, for some observation to be made that is incompatible with the proposition. The statement "all swans are white," for example, could be falsified by observing a green swan. On the other hand, the statement that "there is a green swan somewhere" could only be falsified by observing every swan in existence and noting that none of them are green. For statements like "there is intelligent life elsewhere in the universe," this is at best impractical and may well be considered impossible.
Popper himself proposed falsifiability as a criterion for demarcating what is and is not science. He held, for example, that Freudian psychoanalysis was not "science" because it was not in principle falsifiable; a patient who disagrees with his analyst is simply "in denial," and any contrary argument or behavior on the part of the patient can be ascribed to the patient's attempts to buttress his denial.
Examples of Application
Many philosophical beliefs, including questions of ethics and esthetics, are usually held to be unfalsifiable. A simple example of such a belief is solipsism, the belief that there is no external universe beyond one's own mind. This is obviously unfalsifiable, because whatever "evidence" is presented against it might simply be the product of one's own imagination.
Similarly, questions of ethics (how would you disprove a statement like "murder is evil"?) or esthetics (how would you disprove a statement like "flowers are pretty?") are usually held to be unfalsifiable since they rely on a degree of personal judgement and assumptions.
Statements in mathematics are often unfalsifiable because they are definitions and consequences of definitions; there is little empirical evidence that can be gathered experimentally to touch on question of mathematics.