The obligation of a party to provide sufficient evidence in support of their side of a dispute or issue is called the Burden of Proof.
Actually the term “burden of proof” is a bit strong implying the need to prove beyond a doubt. As a result it is really more like the burden of evidence since the side that has the burden of proof is obligated to provide evidence to back up their view point. However determining who has the burden of proof is not always easy to do because it varies in different circumstances and changes in the course of the discussion.
Under most circumstances the party making the claim has the burden of proof. However this is not an absolute rule given there are some circumstance that can change the burden of proof to the person denying a claim since a party making the new claim about an accepted idea has the burden of proof. For example those claiming that the Apollo Moon landings did not really happen have the Burden of proof.
The Burden of Proof Fallacy is the process of wrongfully trying to switch the burden of proof to your opponent. However the Burden of Proof can legitimately switch sides if new arguments have been made or evidence presented. If the opposition wishes to dispute the new evidence or argument, they have the burden of proof in doing so. Hence the burden of proof has switched.
A common problem in a discussion is agreeing on who has the burden of proof. It can consume much time and render a debate useless. Another problem is that since each side sees the issue differently it may be hard to agree on burden of proof.