In a comment below, Paul offers this definition of moral certainty from St. Alphonsus Liguori:
An opinion of sentiment that is morally certain, is that which excludes all prudent fear of falsity; so that the contrary opinion is regarded as wholly improbable.
I'm happy with that definition myself, but Paul and I don't quite seem to agree on what it means in practice.
I'll first point out that St. Alphonsus speaks of "moral certainty" as applying to an opinion.
St. Thomas describes opinion as a kind of thought about a thing that might be true or false, and he distinguishes opinion from doubt, suspicion, belief, and scientific knowledge. One who opines, he says, "inclines to one side" -- that is, either the side that says the thing is true or the side that says it is false -- "yet with fear of the other."
St. Alphonsus, then, further distinguishes the morally certain opinion as an opinion where the fear of the other side is imprudent. If we don't know a thing is true with scientific rigor -- with "the perfection of clear sight," as St. Thomas might put it -- but our fear of it being false isn't prudent, then we are morally certain of its truth.
Fear is the desire to avoid a potential evil, prudence is right reasoning about a thing to be done, so I'd say that a "prudent fear" is a desire to avoid a potential evil, where the desire is proportionate to both the magnitude of the evil and its potential of occurring.
The point of defining this concept of moral certainty is to be able to say that, for purposes of moral reasoning, a morally certain opinion should be treated the same as a truth known by science or held by faith.