The albatross is symbolic of a few things. As a living bird, the albatross is symbolic of innocence, goodness, God's creation, and even God's love and salvation. In stanza 16 of part 1, the sailors and readers are introduced to the albatross.
At length did cross an albatross,
Thorough the fog it came;
As if it had been a Christian soul,
We hailed it in God's name.
Notice how they hail it as a sign from God. Additionally, it is a good sign because the bird is shown as similar to a Christian soul. That would be something positive and eternal. The bird is beautiful and the sailors see it as a sign of God watching over them. Unfortunately, the Mariner shoots and kills the bird. The dead albatross is then symbolic of sin and bad luck. Many of the hardships that the sailors endure from this point forward are blamed on the Mariner's actions of killing a good creature of God's creation. Coleridge is fairly overt with comparing the killing of the albatross with the crucifixion of Christ. The final stanza of part 2 shows the albatross hanging from the Mariner's neck instead of the cross.
Instead of the cross, the albatross
About my neck was hung.
It isn't until the mariner finally learns to pray that the albatross falls into the ocean and the curse is finally broken, and that is another reminder of how the albatross is symbolically pointing toward faith and God.