Burden of proof is the obligation to prove allegations which are presented in a legal action. It is one of the most important issues in litigation, and in criminal cases it is closely linked with the principle in most modern legal systems that an accused person is innocent until proven guilty. The burden, therefore, initially lies with the plaintiffs in a case, and not on a defendant who would need to prove that something did not happen. Adequate evidence can, however, shift the burden of proof to the other party.
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...