The Oxford Dictionary defines a paradox as follows:
a seemingly absurd or contradictory statement or proposition which when investigated may prove to be well founded or true
mid 16th century (originally denoting a statement contrary to accepted opinion): via late Latin from Greek paradoxon 'contrary (opinion)', neuter adjective used as a noun, from para- 'distinct from' + doxa 'opinion'
In Aha! Gotcha : Paradoxes to Puzzle and Delight Martin Gardner uses the term paradox to include "any result so contrary to common sense and intuition that it invokes an immediate emotion of surprise." He suggests that the four main types of paradox are thus:
The importance of paradoxes should not be underestimated. On the one hand they may simply be a source of entertainment and fascination for the casual observer, but beyond that they play an important role in furthering our understanding of the world around us. They lure us in, infuriate us, and lead to lines of enquiry that might otherwise have been neglected.
Gottlob Frege was a german mathematician who devoted many years of his life to a treatise on the foundations of arithmatic. His work contained frequent referrences to the class of all classes that have a given property. He was just about to publish the second volume (some 10 years after the publication of the first) when he received a communication from Bertand Russell. In this communication Russell outlined what is now know as Russell's Paradox.
Frege responded, "A scientist can hardly meet with anything more undesirable than to have the foundation give way just as the work is finished. In this position I was put by a letter from Mr Bertrand Russell as the work was nearly through the press."