You can't prove a negative
The claim, "you can't prove a negative" (or "...universal negative") is often used as a shorthand in discussions to refer to the difficulty of gathering evidence to "prove" that something does not exist. Proving that a phenomenon is not real takes a lot more time and effort than to demonstrate that it is real. This is especially true when the definition of the phenomenon can be changed at will by its believers (see God). It is 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.
Science, hypotheses and statistics
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 a 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 a well designed clinical trial, that a drug has an effect. However, it is impossible to prove that the drug has no effect: the effect might simply be too small for that particular experiment to detect; a later, larger, or differently designed experiment might well find it. For this reason, scientists and statisticians refer to a failed experiment (one in which the experimental hypothesis was not supported by evidence) as one that "failed to reject the null hypothesis" rather than one that "supported the null hypothesis" (and they never claim that such a result "proved the null hypothesis true").
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 such example is proving a claim which negates a simple, factual untruth. For instance, if one can "prove" that Richard Dawkins is currently in his home in England, then obviously one has proven that Dawkins is not currently in the United States.
Similarly, if any claim implies a logical contradiction, it cannot be true. In the previous example, if one were to claim that Dawkins was in England and in the United States at the same time, then the claim itself would be a contradiction. As an example of a claim that implies a contradiction, consider this mathematical statement: "There are no prime numbers whose square root is a rational number." This is a "universal negative" that is relatively easy to prove using a "proof by contradiction" (see Wikipedia:Irrational number), which is a form of reductio ad absurdum. See the latter article for more examples.
Reductio ad absurdum is a form of modus tollens argument. 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.