Special pleading is a logical fallacy in which one violates the Principle of Relevant Difference. In this fallacy, one acknowledges a general rule but asserts that there is an exception to the rule, without adequately defending that the exception is valid.
Special Pleading: Yes, murderers should go to jail, but John was really a serious jerk; I should be excused for killing him.
NOT special pleading: Yes, murderers should go to jail, but our jurisdiction allows self-defense as long as the response matches the threat, and the CCTV footage will clearly show that John was aiming a gun at my head; I should be excused from killing him.
Special pleading can be quite hard to spot. Consider this argument:
"God could not have created the world just 6000 years ago, because the furthest stars we can see are over a million light years away. If the universe hadn't been around for at least that long, we wouldn't be able to see them."
In this case the person making the argument is (at least provisionally) taking the existence of God as a hypothesis under the argument, and allowing that the idea of God creating the entire universe is valid, but is making a special plea that God could or would not place photons in certain places. What initially appeared to be an intelligent argument quickly becomes ridiculous, and trivial, if we make the plea explicit: "If you believe in Recent Creation and a bunch of technical scientific measurements, then you would have to believe that God is willing and able to place photons in certain places".
Despite my example, the people most often accused of special pleading are fundamentalists and the religious right, who have long-standing cases of special pleading ranging from an uncaused first cause to a freedom to eat pork. This is probably primarily due to 'special pleading' being a popular skeptic meme rather than it being a particular fault of religious conservatives, but it worth noting that any claim of god(s), the supernatural, or spirituality may be seen as a form of special pleading when used in a scientific debate -- and vice versa.
Related fallacies include: One-sided argument, Moving the goalposts, Proof by handwaving, Tu quoque, and No true Scotsman.