The concept of a burden of proof is important in debates whoever has a burden of proof is obligated to prove their claims in some fashion. If someone doesnt have a burden of proof, then their job is much easier: all that is required is to either accept the claims or point out where they are inadequately supported.
It is thus no surprise that many debates, including those between atheists and theists, involve secondary discussions over who has the burden of proof and why. When people are unable to reach some sort of agreement on that issue, it can be very difficult for the rest of the debate to accomplish much. Therefore, it is often a good idea to try to define in advance who has the burden of proof.
The first thing to keep in mind is that the phrase burden of proof is a bit more extreme than what is often needed in reality. Using that phrase makes it sound like a person has to definitely prove, beyond a doubt, that something is true; that, however, is only rarely the case. A more accurate label would be a burden of support the key is that a person must support what they are saying. This can involve empirical evidence, logical arguments, and even positive proof.
Which of those must be presented will depend very much upon the nature of the claim in question. Some claims are easier and simpler to support than others but regardless, a claim without any support is not one which merits rational belief. Thus, anyone making a claim which they consider rational and which they expect others to accept must provide some support.
An even more basic principle to remember here is that some burden of proof always lies with the person who is making a claim, not the person who is hearing the claim and who may not initially believe it. In practice, then, this means that the initial burden of proof lies with the theist, not with the atheist. Both the atheist and the theist probably agree on a great many things, but it is the theist who asserts the further belief in the existence of a god.
This extra claim is what must be supported, and the requirement of rational, logical support for a claim is very important. The methodology of skepticism, critical thinking, and logical arguments is what allows us to separate sense from nonsense; when a person abandons that methodology, they abandon any pretense of trying to make sense or engage in a sensible discussion.
The principle that the claimant has the initial burden of proof is often violated, however, and it isnt unusual to find someone saying, Well, if you dont believe me then prove me wrong, as if the lack of such proof automatically confers credibility on the original assertion. Yet that simply isnt true indeed, its a fallacy commonly known as Shifting the Burden of Proof. If a person claims something, they are obligated to support it and no one is obligated to prove them wrong.
If a claimant cannot provide that support, then the default position of disbelief is justified. We can see this principle expressed in the United States justice system where accused criminals are innocent until proven guilty (innocence is the default position) and the prosecutor has the burden of proving the criminal claims.
Technically, the defense in a criminal case doesnt have to do anything and occasionally, when the prosecution does an especially bad job, you will find defense lawyers who rest their case without calling any witnesses because they find it unnecessary. Support for the prosecution claims here is so obviously weak that a counter-argument simply isnt deemed important.
In reality, however, that rarely happens. Most of the time, those required to support their claims do offer something and then what? At that point the burden of proof shifts to the defense. Those who do not accept the support offered must at the very least show just cause why that support is insufficient to warrant rational belief. This may involve nothing more than poking holes in what has been said (something defense attorneys often do), but it is often wise to construct a sound counter-argument which explains evidence better than the initial claim does (this is where the defense attorney mounts and actual case).
Regardless of exactly how the response is structured, what is important to remember here is that some response is expected. The burden of proof is not something static which one party must always carry; rather, it is something which legitimately shifts during the course of a debate as arguments and counter-arguments are made. You are, of course, under no obligation to accept any particular claim as true, but if you insist that a claim isnt reasonable or credible, you should be willing to explain how and why.