Definition
Burden of proof is the obligation to support a claim with sufficient evidence and reasoning before it counts as established. The burden rests on whoever makes the claim, not on whoever doubts it. In a formal debate, the side proposing a new policy or making an affirmative case typically carries the burden of proving that case; the responding side does not have to disprove it from nothing, only to show the case as presented has not met its burden, unless they choose to make their own affirmative claims.
Example
If a proposition team claims a new policy will reduce crime, the burden is on them to show a plausible mechanism and supporting evidence. It is not the opposition's job to prove crime will stay the same; it is enough for them to show that the proposition's mechanism does not hold up, or that the evidence offered does not actually support the claim made.
Common mistakes
A frequent error is "shifting the burden": demanding that an opponent disprove an assertion that was never itself supported. "You haven't shown my plan won't work" is not a valid response to a claim offered with no evidence behind it in the first place. If a claim was not supported, the correct response is to note that it has not met its burden, not to take on the work of disproving it yourself.