These are both legal burdens of proof. In a criminal case, the state must prove all elements of the offense beyond a reasonable doubt. In a civil case, the claimant must prove their allegation by the preponderance of the evidence. You can think of this as a balance of probabilities. If the claimant proves there case such that it is more likely than not (>50%), they win.
If you are a defendant in say Nashua District court for a misdemeanor offense, the state has the burden to prove each element of the offense beyond a reasonable doubt. This is a good thing because it is the highest legal standard in our legal system.