Perhaps I should say . . . the unbaptized sinner?

As the Roman Catholic Church understands it, did the fall of Adam completely erase the image of God in man so that man is nothing more than Augustine's "massa damnata?"

If the answer is yes, then what is the image of God? I have heard that it is the ability to choose (some may call it "free will.")

We had free will before the fall, and we have free will now. That was not lost. Being made in the image & likeness encompasses our free will and having a rational soul (one that can experience the supernatural).

We originally possessed:
perfect human will
perfect human intellect
perfect grace in our soul

What we have been left with now:
weakened will
darkened intellect
fallen nature

CCC 405
It is a deprivation of original holiness and justice, but human nature has not been totally corrupted: it is wounded in the natural powers proper to it, subject to ignorance, suffering and the dominion of death, and inclined to sin - an inclination to evil that is called "concupiscence". Baptism, by imparting the life of Christ's grace, erases original sin and turns a man back towards God, but the consequences for nature, weakened and inclined to evil, persist in man and summon him to spiritual battle.

Yes, but is that the image of God?

What happened to the image of God when we were conceived? Is it present or was it totally erased by Adam's sin?

Neither. It is present, but it's damaged.