Harry Houdini was a famous illusionist, magician, and escape artist of the 19th and 20th century. He was famous for his ability to release himself from any restraint. Some believed he was double-jointed and that this explained his skill with escapology, though the vast majority of his exceptional abilities were likely provided by both a large selection of carefully hidden lock picks and extensive physical training.
Captain Jonathan Archer described Houdini's ability to T'Pol when two of them were captured and bound together by Coridanite dissidents in 2151. T'Pol stated Houdini's unique ability seemed "unlikely." (ENT: "Shadows of P'Jem")
In the first draft script of "Shadows of P'Jem" (written while the episode had the working title "Untitled Andorians Return"), Houdini's first name wasn't given, though it is said in the final version of that episode, and Archer admitted that he had never been one of the people who believed Houdini was double-jointed.
Commander Trip Tucker compared Hoshi Sato to Houdini in 2154, after Sato managed to free herself from a locked decon chamber. (ENT: "Observer Effect")
During the Dominion War, Starfleet personnel assigned to AR-558 called subspace mines that the Dominion used against them "houdinis", due to their ability to appear and disappear. (DS9: "The Siege of AR-558")
According to the Star Trek Encyclopedia, 4th ed., vol. 1, p. 348, Houdini spent his later career to debunking psychics and mediums.