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.

Coleridge's "The Rime of the Ancient Mariner" features a protagonist that loses his former self and his shipmates due to a casually careless and cruel act, and is led to regeneration by his losses.

The Mariner is not intentionally cruel.  He kills the albatross on a whim.  He just doesn't respect nature as part of existence and God's creation.  Natural forces, in the form of the supernatural, teach him a lesson.  By the time he accosts the wedding guest he knows better than to disrespect nature.  His purpose, in fact, in cornering people and telling them his story is to convince them to respect all of God's creatures.  He suffers loss and is led to regeneration. 

The wedding guest, too, undergoes a bit of a transformation and regeneration.  He is dismissive of and disrespectful to the Mariner at the beginning of the poem, but subdued and passive after he is forced to listen to the tale.  Thus, the possibility of regeneration does exist--the Mariner and the guest demonstrate this. 

The Mariner's intent, his message, is in some ways an old one, of course.  Yet Coleridge avoids being didactic (preachy) by altering the traditional Christian message from faith and following God to loving and respecting nature.  Humans need to repent their lack of respect of all of nature, not their lack of faith in God.  He also avoids being didactic by creating a work of outlandish imagination, full of supernatural beings.