Shifting the burden of proof

Shifting the burden of proof is a kind of logical fallacy in argumentation whereby the person who would ordinarily have the burden of proof in an argument attempts to switch that burden to the other person, e.g.:
 * If you don't think that the Invisible Pink Unicorn exists, then prove it!

Establishing the burden of proof
In an argument, the burden of proof is on the person making an assertion. That is, if a person says that the moon is made of cheese, then it is up to that person to support this assertion. Demanding that the other party demonstrate that the moon is not made of cheese would constitute shifting the burden of proof.

It can sometimes be tricky to determine who holds the burden of proof for a given assertion. If a bank's customer sues the bank, claiming that money was illegally withdrawn from his account, the customer is making a positive assertion, and should therefore have the burden of proving it. However, the bank has detailed financial records, and it is therefore easier for the bank to demonstrate that nothing illegal occurred, than it is for the customer to demonstrate that something illegal happened.

Secondly, a rebuttal to a positive claim can also be a positive claim, which also must be supported by evidence. Imagine the following conversation:
 * A: The moon is made of green cheese.
 * B: That's not true: astronauts have gone to the moon and found that it's made of rock.

Here, the statement "the moon is made of green cheese" is a positive assertion, and person A has the burden of proving it. The second statement is a rebuttal of the first, but the statement "astronauts have gone to the moon" is a positive assertion. Since B is making a positive assertion (about space travel), it is B who has the burden of proving it.

Atheism
A theist is someone who claims that there is a god. An atheist is someone who doesn't. Since the theist is the one making a positive claim, it is the theist's job to demonstrate that a god exists.

Well-established scientific theories
Often, in debates over well-established scientific theories, the person arguing against the mainstream view will say that it is not up to him to disprove the theory, but that it is scientists' job to demonstrate it.

This is true, in a sense: when a new scientific hypothesis is introduced, its proponents have the onus of demonstrating it. The rest of the scientific establishment has no obligation to disprove the new hypothesis. However, a hypothesis can only rise to the rank of theory by being repeatedly tested, and by accumulating evidence in its favor. This evidence must now be taken into account by the theory's critics.

Thus, if person A says that relativity is unproven, and person B asks A for evidence, this may be seen as shifting the burden of proof, but B is really asking A to support the positive assertion that the mass of evidence for relativity is not conclusive.

In online debates, when a person challenges a well-established scientific theory, it is almost invariably the case that that person does not know or does not understand the evidence for the theory.