The living albatross is a symbol of God's creation and of innocence. The dead albatross is a symbol of sin.
When the Mariner kills the albatross, the other sailors see this as a sign of bad luck and fear, rightfully, that their dangerous voyage will be cursed and run into trouble. Yet the "bad luck" is merely a surface manifestation of a deeper issue. God created the albatross and all of the earth's creatures, and thus they all become reminders of God's presence in the world. They are not merely objects. The albatross is not simply a bird but a carrier of the divine spark. Albatross flesh supposedly tasted like human flesh, highlighting that both albatross and human are touched with divinity.
It is only when the Mariner can feel the beauty of God's creation that the curse of killing the albatross is lifted. When he feels love for the water snakes, the Mariner is released from his sin, and the albatross falls from his neck.