The Question and Answer section for The Rime of the Ancient Mariner is a great resource to ask questions, find answers, and discuss the novel.
On the surface of things, the Mariner kills the albatross because he blamed the lack of wind on the bird. The men were petty, as illustrated through their reactions. When the wind blew, it was all "good albatross", but when the wind stopped, it...
This is actually a pretty detailed question for this short space. Fortunately it is answered under "Religion" at the GradeSaver link below:
The mariner shot and killed the albatross.