Special pleading is a claim that standards of evidence should be modified or reversed for a particular claim or type of claim.
For example, while the standard practice for analyzing a claim places the burden of proof on the individual making the claim, some religionists seek to reverse this burden, asserting that belief in the existence of a god is warranted until the evidence and arguments opposing this claim are sufficient to warrant disbelief.
Another good example of special pleading was done by Charles Oxnard in a study on Australopithecines. In it he claimed that his results showed that australopithecines were no more closely related to humans than apes. The study was critized for several reason and is not widely accepted in the scientific world. However, the study is widely quoted by Creationists such as Duane Gish because of Oxnard's use of computers in his research. Gish claims that since "[a] computer doesn't lie, [a] computer doesn't have a bias" Oxnard's should be taken seriously. This is where the special pleading comes in; a great deal of studies have been done using computers that contradict Oxnard's results yet Creationists don't want us to take them seriously. This is also an example of cherry picking; counting the times that computers have backed up Creationism yet forgetting the times they haven't.
Creationist Arguments: Australopithecines at talkorigins.org