You can't prove a negative
The claim "you can't prove a negative" is often used as a shorthand in discussions to refer to the difficulty of gathering experimental evidence to "prove" that something does not exist. Proving that a phenomenon isn't real takes a lot more time and effort than it takes to demonstrate it. This is especially true when the definition of the phenomenon can be changed at will by its believers. Its very difficult to prove the general non-existence of a phenomenon, and this difficulty is used by believers of many kinds of phenomena to give the appearance of credibility to their beliefs.
Much scientific practice has developed to address this issue. In particular, the field of statistics distinguishes between the so-called experimental hypothesis and the null hypothesis. The experimental hypothesis is usually the statement that the scientist would like to investigate the truth of (for example, that the drug under study is an effective treatment), while the null hypothesis is the opposite (that the drug is ineffective). It is possible to prove, by gathering a clinical group together, that the drug has an effect -- but it is impossible to prove that the drug has no effect; it might happen that the drug has an effect, but one too small for that particular experiment to notice (and that a later, larger, or differently run experiment might find it). For this reason, scientists and statisticians refer to a failed experiment as one that "failed to reject the null hypothesis" -- one where all the evidence available was negative, but the null hypothesis is still not "proven".
Because it is overly broad, this phrase is often overused or misapplied. Contrary to the claim, it can be just as easy to prove a negative as a positive.
One example of a situation where one can prove a negative is for a claim which negates a simple factual untruth. For instance, to the extent that it is possible to "prove" that "Richard Dawkins lives in England," it is then trivial to prove that "Richard Dawkins does not live in the United States."
Similarly, any claim that implies a logical contradiction cannot be true, and thus you can prove that it is not true. For example: "I know a man who both is exactly six feet tall, and is not exactly six feet tall." This statement is not true for any person, and one need not be omniscient to assert that there is no such man.
Although a claim may not directly contradict itself, it is a common practice in argumentation to show that the claim leads to an inevitable contradiction. This technique is known as modus tolens, or reductio ad absurdum.
Strong atheists who assert that there is no god may sometimes rely on this tactic, for instance by invoking the argument from evil to show that a god with some set of characteristics cannot exist in the known world.