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.