A fairy is a mythical creature that has taken on many different forms in the popular imagination throughout history. The most common modern conception, which became very popular in the early 20th century, is more or less like a more elegant (usually somewhat angelic) version of the Peter Pan character Tinkerbell — a small, flying, female humanoid. In the past, however, the term has referred to a variety of magical creatures, such as trolls.
The significance of fairies to counter-apologetics is that they can be used in arguments as substitutes for God (or angels, etc.) in arguments about the reasonableness of belief without evidence. Since some people still actually do believe in fairies, however, sometimes better use can be made of other mythical creatures, such as leprechauns or unicorns (see the latter article for an example).