A fee paid for a completed article done on assignment subsequently not published for a variety of reasons. The amount is usually 25-50% of what would have been paid if the piece had been published. Some publishers will pay 100% of the original scheduled payment. The amount of the kill fee, if needed, should be established prior to writing the article, typically in a contract. Keep in mind when a publisher responds positively to a query and tells you to send in your manuscript, they are looking at it on speculation. That is not considered an assignment. It simply means they are willing to take a look at it and if they like it will offer to buy it. If it does not meet their needs after all, they will simple return or discard it.
Usually an editor will not make a definite assignment unless the writer has written for that publication in the past—so the editor knows the writer's work or reputation—and can be quite sure the writer will present them with a publishable piece. It would be unusual for a new writer to be given an assignment unless the topic was irresistible. In that case they may accept it but have to do a considerable amount of editing to make it publishable. For a more in-depth look at kill fees, go to: http://www.writersdigest.com/editor-blogs/questions-and-quandaries/legal-questions/what-is-a-kill-fee.