Burden of proof is essentially the amount of proof needed to prove something. The amount of proof changes depending on the type of case. For example, to convict someone of a criminal offense and strip them of their freedom, you must have proof “beyond a reasonable doubt.” This is the most proof required in any legal case—as it should be.
On the other hand, for a civil case where money is at issue, you need a “preponderance” of the evidence. This is essentially enough evidence to tip the scales of justice only slightly. Put another way, you will have a preponderance of the evidence if you have 51% of the evidence and the other side has 49%.
In between criminal and civil cases is “clear and convincing evidence”. This is the amount of proof needed in child custody cases. It’s less than the proof needed to take someone’s freedom away, but more than needed to win a money judgment against someone.