What is the dominant theory of atonement in Catholic theology? Having a little trouble phrasing the question due to my own ignorance. 
I grew up celebrating Christmas and Hanukah as basically family gift giving events, as many others do, and I'm trying in adulthood to have at least a basic understanding of actual religious doctrine. 

In general I'm trying to receive an answer from someone who can explain why Catholicism follows a particular theory of atonement over others, and in doing so elucidate as to how these theories differ at least to the degree that one is more favourable for Catholics over another or others. 

Looking for a simple explanation, hopefully.

From my (limited) understanding there are the following broad theories:
Satisfaction theory of atonement, substitutionary atonement, penal substitution, ransom theory, maybe more.
There is no single theory of atonement in Catholicism, but rather the doctrine has developed over time due to speculation from various theologians. Though out of the four you've provided, satisfaction theory is the only one I think you will find an orthodox catholic espousing. The Catholic Encyclopedia entry on atonement is helpful.

it may be noted that in this instance the development is chiefly due to Catholic speculation on the mystery, and not, as in the case of other doctrines, to controversy with heretics. At first we have the central fact made known in the Apostolic preaching, that mankind was fallen and was raised up and redeemed from sin by the blood of Christ. But it remained for the pious speculation of Fathers and theologians to enter into the meaning of this great truth, to inquire into the state of fallen man, and to ask how Christ accomplished His work of Redemption (emphasis mine). By whatever names or figures it may be described, that work is the reversal of the Fall, the blotting out of sin, the deliverance from bondage, the reconciliation of mankind with God. And it is brought to pass by the Incarnation, by the life, the sufferings, and the death of the Divine Redeemer. All this may be summed up in the word Atonement. This, is so to say, the starting point. And herein all are indeed at one. But, when it was attempted to give a more precise account of the nature of the Redemption and the manner of its accomplishment, theological speculation took different courses, some of which were suggested by the various names and figures under which this ineffable mystery is adumbrated in Holy Scripture. Without pretending to give a full history of the discussions, we may briefly indicate some of the main lines on which the doctrine was developed, and touch on the more important theories put forward in explanation of the Atonement.

The page explains that substitutionary theories of atonement (and here I include as well penal substitution) are a product of post-reformation protestantism. So we don't follow those theories because our theologians never put them forward.

And in their explanation of the merit of Christ's sufferings and death we may see the influence of St. Thomas and the other great Schoolmen. But, as might be expected from the isolation of the doctrine and the loss of other portions of Catholic teaching, the truth thus preserved was sometimes insensibly obscured or distorted. It will be enough to note here the presence of two mistaken tendencies.

The first is indicated in the above words of Pattison in which the Atonement is specially connected with the thought of the wrath of God. It is true of course that sin incurs the anger of the Just Judge, and that this is averted when the debt due to Divine Justice is paid by satisfaction. But it must not be thought that God is only moved to mercy and reconciled to us as a result of this satisfaction. This false conception of the Reconciliation is expressly rejected by St. Augustine (In Joannem, Tract. cx, section 6). God's merciful love is the cause, not the result of that satisfaction.

The second mistake is the tendency to treat the Passion of Christ as being literally a case of vicarious punishment (emphasis mine). This is at best a distorted view of the truth that His Atoning Sacrifice took the place of our punishment, and that He took upon Himself the sufferings and death that were due to our sins.



Any sort of ransom theory also seems unreasonable and has been met with unanswerable arguments from Anselm and others:

But however useful and illuminating in their proper place, figures of this kind are perilous in the hands of those who press them too far, and forget that they are figures. This is what happened here. When a captive is ransomed the price is naturally paid to the conqueror by whom he is held in bondage. Hence, if this figure were taken and interpreted literally in all its details, it would seem that the price of man's ransom must be paid to Satan. The notion is certainly startling, if not revolting. Even if brave reasons pointed in this direction, we might well shrink from drawing the conclusion. And this is in fact so far from being the case that it seems hard to find any rational explanation of such a payment, or any right on which it could be founded... This curious notion, apparently first mooted by St. Irenæus, was taken up by Origen in the next century, and for about a thousand years it played a conspicuous part in the history of theology... A protest was raised by St. Gregory of Nazianzus in the fourth century, as might be expected from that most accurate of the patristic theologians. But it was not till St. Anselm and Abelard had met it with unanswerable arguments that its power was finally broken. It makes a belated appearance in the pages of Peter Lombard.