The answer to the question about the Ancient Mariner's atonement is in Part Seven of "The Rime of the Ancient Mariner," starting around Stanza 11. When The Mariner has come safely to his own "countree," and has been given a beacon by the "seraph-band" ("A man all light, a seraph-man, / On every corse there stood"), and has driven the Pilot's boy mad and heard the Hermit's plea to "shrieve" him, his atonement becomes apparent.
The Mariner says in Part Seven that at the Hermit's pleading and questioning, his "frame" (body) was "wrenched / With woful agony." This overpowering agony compelled him, forced him, to tell the tale of his ill-begotten voyage to the listener, in this case the Hermit.
The Mariner then says that ever after that first telling of the tale to the Hermit, he wanders the land "like night," dark and invisible, and possesses a "strange power of speech," a power that transfixes his listener. He also says that, in the midst of his wanderings, from time to time "at an uncertain hour" (an unknowable hour), he'll see a face and burn with the same agony that forces him to tell his tale anew to the face of the "man that must hear" his grisly tale.
In Coleridge's "The Rime of the Ancient Mariner," the mariner must tell his tale any and every time he sees a man's face and feels the compulsion to tell that specific man. You can find this in lines 578-590 of the poem in Part VII.
The idea is that to atone for his sin the man must spread the message that humans should love all of God's creatures. As the mariner says a few stanzas later:
"He prayeth best, who loveth best
All things both great and small;
For the dear God who loveth us,
He made and loveth all."
He tells the tale for the first time to the Hermit who rescues him from the sea, and from that point on he must tell it any time he feels the compulsion to do so.