The preponderance of evidence is significantly related to the notion of the burden of proof, and it refers to which side of the trial needs the majority of the evidence in order to prove its point.
By majority of evidence in the prior sentence, what is meant is the importance of evidence, the accuracy of the evidence, and the convincing nature of the evidence, as opposed to pure quantity of evidence. A preponderance of evidence thus might be generated from a single witness who provides reliable, clear, and truthful testimony which invalidates all the numerous points of evidence generated by the other side of the trial.
In a civil trial, in order for either side to be considered to have successfully proved its case, it must generate a preponderance of the evidence for its side. This means that its evidence must ultimately outweigh the evidence of the other side, and fits into the conception that the jury ultimately picks the better argument for the trial.
This is significantly different, however, from what is involved in a criminal trial, where the burden of proof is much stronger upon the prosecution. In a criminal trial, the case must be proved beyond a reasonable doubt, as opposed to simply with a preponderance of the evidence, meaning that if the prosecution makes an argument which is better than the argument of the other side, but does not prove beyond a reasonable doubt that the defendant is guilty, then the prosecution will not win the case. A preponderance of the evidence is thus much more easily achieved.