The obligation to prove one's case.
The burden of proof is always on the person making an assertion or proposition. Shifting the burden of proof, a special case of argumentum ad ignorantium, is the fallacy of putting the burden of proof on the person who denies or questions the assertion being made. The source of the fallacy is the assumption that something is true unless proven otherwise.
From Black's Law Dictionary:
The obligation of a party to establish by evidence a requisite degree of belief concerning a fact in the mind
of the decision-maker in order for that party to prevail that fact.
In anglo-american law, the traditional notion is that someone seeking something has the burden of demonstrating why they are entitled to it. For example, if person A is being prosecuted for murder by court B, then it is up to B to prove beyond reasonable doubt that A perpetrated the act.