Abarim Publications' online Biblical Greek Dictionary
The noun φιαλη (phiale), hence our English word vial, describes a dish or flat bowl with a signature wide open mouth. It's unclear where this word may have come from, but the -αλη (-ale) part is a common suffix for instruments. The first part of our word, for lack of better suggestions, may very well stem from the Hebrew noun פה (peh), meaning mouth.
Our noun φιαλη (phiale) occurs 12 times in the New Testament (see full concordance), all in the Book of Revelation, where vials are filled with either the prayers of the saints or else with various manifestations of the wrath of God that is poured out onto the earth. Neither saintly prayers nor the wrath of God are physical liquids, so these vials are not physical vials. What they are, then, is not immediately clear, but it's striking that the author didn't describe bottles but rather flat dishes, and so does not emphasize the final pouring but rather the initial collecting.
Here at Abarim Publications we don't know either, but with God, all bills must always be paid (which is why his Son had to die: to make a restitution for a debt that could neither be ignored nor expunged in some other way). The same principle demands that when we create order somewhere, we must pay for this order by an increase in disorder somewhere else — this is the second law of thermodynamics: in a closed system, general entropy must always increase, so a local reduction in entropy (say, an art work on your desk) must be paid for by an increase in entropy around that locality (around your desk).
Here at Abarim Publications we surmise that the prayers of the saints somehow counteract the effects of lawlessness in the world, whereas the wrath of God arises from an illegitimate creation of unnatural order — say, big ugly concrete buildings with people in them who do things that go against all of their hearts of hearts; people who tyrannize populations with nonsense legislation or harsh taxes, and who form the world at large according to their own unholy financial image.