In common usage, a canard is a deliberately false or ungrounded story, normally used to back up an otherwise unfounded argument. Examples of this include the contention that Jews faked the Holocaust, George Bush deliberately planned 9/11, or that the media is involved in a left-wing conspiracy. Quite appropriately, quacks love using them.
In anti-science and hard-right conservative usage, illustrated on sites such as WorldNetDaily and Free Republic, a canard is any frequently used argument that damages their credibility, and for which one has no credible response, or is unwilling to attempt to counter. Examples of this usage include pre-Cambrian rabbits not falsifying evolution. Another classic conservative canard is that "the government can never do anything as well or as efficiently as private concerns can."
An example of a liberal canard is that independently-funded research is less subject to conflict of interest than corporate-funded research, without giving evidence or reasons why a given piece of research is suspect.
"Canard" happens to be French for duck. In aircraft design, it also refers to a type of control surface mounted in front of the wing (as opposed to conventional horizontal stabilizers situated behind the wings), which someone a long time ago found ducklike. (European aircraft manufacturers such as Dassault and Saab, as well as fans of Burt Rutan's plane designs, are particularly fond of them, and Russian aircraft manufacturer Sukhoi decided to use both canards and horizontal stabilizers in one design.)
"Mon canard" (literally: "My duck") is a French term of endearment. The word "canard" in French is also an old slang word meaning "newspaper".