Meaning:- A burden which some unfortunate person has to carry.
This phrase refers to lines from the poem The Rime of the Ancient Mariner by Samuel Taylor Coleridge, in which the eponymous mariner, who shoots an albatross, is obliged to carry the burden of the bird hung around his neck, as a punishment for and reminder of his ill deed.
Coleridge published the work in 1798, in the collection of poems that is generally accepted as being the starting point of the Romantic movement in English literature - Lyrical Ballads, with a Few Other Poems:
The Rime of the Ancient Mariner
God save thee, ancient Mariner
From the fiends, that plague thee thus
Why look'st thou so ? - With my cross-bow
I shot the ALBATROSS.
Ah. well a-day. what evil looks
Had I from old and young
Instead of the cross, the Albatross
About my neck was hung.~Samuel Taylor Coleridge