An abiding conviction is a very strong belief of a defendant's guilt based on the evidence presented. The terms “abiding conviction” or “to a moral certainty,” are commonly used when instructing juries on the frame of mind necessary to find that a defendant's guilt has been proved beyond a reasonable doubt, which is the burden of truth in a criminal trial. In other words, the jury in a criminal trial will be given jury instructions that help them determine whether each jury members' belief of the defendant's guilt is strong enough to meet the “beyond a reasonable doubt” standard.
It is a stronger burden of proof than the one used in civil trials, which is a “preponderance of the evidence”. For example, if during jury deliberations, juror #11 is wavering on the issue of whether the prosecution has proved the defendant's guilt (e.g., she can see it both ways), juror #11 does not have an abiding conviction of the defendant's guilt and could not vote for conviction.