"You can't prove a negative."
This statement gets tossed around a lot in arguments, especially arguments that lend themselves to the more abstract and theoretical planes, like "Does God exist?" or "Are we alone in the universe?" In those contexts, it's perfectly true --- you can't establish, once and for all, that X (whatever elusive entity X might be) doesn't exist.
A similar precept goes, "Absence of evidence is not evidence of absence."
But if it's true that you can't prove a negative, why is science news so regularly coming out with stories announcing such proofs? "Danish researchers find no link between thimerosal and autism," say, or "Mozart's music does not make you smarter," or "[G]iving up caffeine does not relieve tinnitus" --- all of these statements imply that something has been shown not to be the case.
I think the difference between those statements and the kind of "proving a negative" that's supposed to be impossible lies in how the questions are phrased. For a hypothesis to be testable, it has to have a set of conditions that must also be true --- and are measurable --- if the hypothesis is true. If you can make a prediction based on your hypothesis --- say, that if it's true that thimerosal in vaccines causes autism in children, then autism rates among school-aged children should fall as people stop using thimerosal in vaccines --- and if that prediction's failing to come true necessarily means your hypothesis was wrong, you can falsify the hypothesis. Based on what happens, your hypothesis can be proved wrong or right.
There's nothing inherently impossible about proving that a hypothesis is not true; usually, the statements that cannot be disproven are so vague, or deal with such a vast array of possibilities (i.e., "Space aliens exist somewhere in the universe") that there's no way to test them.