How the bird had played an important role in the poem

Are you referring to the albatross?

The albatross was a symbol of good luck and fair wind. The sailors felt that the Mariner destroyed any good fortune they will have.

And I had done a hellish thing,

And it would work 'em woe:

For all averred, I had killed the bird

That made the breeeze to blow.

Ah wretch! said they, the bird to slay,

That made the breeze to blow!

The Mariner has to atone for his sin of killing the albatross.