Burden of proof
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"Burden of proof" is the obligation that somebody presenting a new or remarkable idea has to provide evidence to support it. In a scientific context evidence is experimental or empirical data (although in some branches, well thought out mathematics may suffice). Once some evidence has been presented, it is up to the opposing "side" to disprove the evidence presented or explain why it may not be adequate. For example, in identifying a chemical compound, an analyst may present a spectrum to support their hypothesis but a reviewer may point out that it is insufficient, explain why by offering an alternative interpretation and state more data is needed, usually suggesting specific data that would be required. This sort of procedure happens constantly in the scientific method, repeating until everyone is happy that the data and explanation match.
Denialists of evolution and global warming have a habit of merely ignoring evidence and claiming that the burden of proof still rests with the proponents of those concepts. Debates frequently descend into arguments about which side the burden of proof lies with.
Often, someone will present a new idea and say that it must be accepted because it cannot be disproved. This is insufficient because without evidence there is no reason to accept an idea, even if there is no contrary evidence. One example is that of a simulated reality, which proposes that the human race does indeed live in The Matrix and we are a computer simulation. There is no evidence against this idea, in fact, it may be impossible to fully disprove, but as there is no real evidence for it there is no reason to accept the idea as real. Another famous example is the teapot proposed by Bertrand Russell, the existence of which cannot be disproved.
Indeed, if one is making an exceptionally bold claim (such as with alternative medicine that claims miraculous cures) then exceptional evidence is expected in its support. Or as Carl Sagan said, "extraordinary claims require extraordinary evidence".
 Null hypothesis
If someone has presented you with an idea and says that the burden of proof is on you to disprove the idea, work out what the null hypothesis is and then put their evidence for the idea against it.
The person claiming something is possible or has happened needs to produce evidence to refute the null hypothesis.
If they have considerable and well-tested evidence, the burden of proof may reasonably be considered to be on the person claiming that the evidence does not hold.
Theists would like to maintain that the existence of a god, or even many gods, is somehow controversial in terms of the burden of proof. It is clear that gods are objects/persons/entities that have been proposed to exist, therefore the burden of proof is on those who claim one exists.
Believers, however, are so sure of the existence of their gods (or, perhaps, fairies) that they claim that those who do not believe in their gods must somehow "prove" their god's non-existence - even in the absence of any evidence for them.