The Mariner violates God's will when he kills the albatross. He has sinned against God, because God, who created everything, loves all of his creation.
The Mariner is not a bad person. He simply does not have a well-developed imagination or sense of empathy. When he kills the albatross, it never occurs to him that he is killing a beautiful creature that God created, one that deserved to be treated with tenderness and respect.
Until the Mariner can understand what he has done, he is cursed, along with his shipmates. When is able to perceive the beauty and marvel of the sea snakes, he has learned his lesson and can be regenerated, or, in other words, freed from his curse.
So, yes, the poem is based on the concept of sin and regeneration. But Coleridge interprets these traditional Christian concepts through the lens of Romanticism, which highly prized nature as the manifestation of God on earth (not as God, but as evidence of God's presence). Man's dominion over the earth, does not, in Coleridge's eyes, allow mankind to indiscriminately use nature for its own pleasure, which is how some Christians interpreted the Bible. Instead, human beings are to love and care for the natural world.