In practice, the question of who has burden of proof usually does not change the outcome of a case, because prosecutors rarely bring cases which are marginal enough so that who has burden of proof makes a difference. In any case, most criminal cases in the United States are resolved via plea bargaining in which burden of proof again does not make a significant difference to the outcome of the case.
Outside a legal context, "burden of proof" means that someone suggesting a new theory or stating a claim must provide evidence to support it: it is not sufficient to say "you can't disprove this".
For example, if I say "The Chinese government is plotting to poision our water supply" it is my burden of proof to prove this plot is actually occurring. If I can provide evidence that proves the plot exists, then it becomes the skeptic's burden of proof to disaprove my claim with facts of his own.
A classic example comes from Criswell's final speech at the end of Ed Wood's Plan 9 from Outer Space: "My friends, you have seen this incident, based on sworn testimony. Can you prove that it didn't happen?". Remember that we're talking here about grave robbers from space...