The Philadelphia Experiment allegedly occured in 1943, during the height of World War II. During this period the Americans had not yet landed in Normandy and were viciously fighting Germany U-Boats in the Atlantia. This was the height of the Battle of the Atlantic. Allegedly the Americans fitted a ship with a cloaking device, using Albert Einstein's Unifed Field Theory notations. What was supposed to happen was that the ship would be able to cloak itself and the shipping freighters so German U-boats were unable to torpedo them.
The ship in question was the USS Eldridge and the ship itself was in Philadelphia at the time. However, many consider that this experiment was a hoax because there was only one eye-witness, that of Carl Allen. Allen claims that the ship disappeared, reapperaed in the Norfolk shipyards, then teleported back to Philadelphia. Witnesses say that the ship disappeared in a haze of greenish smog and that when it reappeared some of the sailors experience extreme nausia and that others had been binded to the metal structures of the ship. Limbs were hanging out of metal walls, some sailors were cut in two, some had vanished completely.
There are many inconsistancies with the proposed experiment. First the Navy has completely debunked it and claimed that it was simply "science fiction" others have stated based on records that the USS Eldridge was in New York harbor during the proposed dates of the exerimentation and never made dock in Philadelphia. With the bulk of the eyewitness account coming from one sailor Carl Allen, it seems like the Philadelphia Experiment, while fantastical, is simply a hoax.
The Experiment has spun numerous webs in Science Fiction video games. Games such as Half Life 2: Episode 2 show a ship named the Borealis disappearing with a piece of drydock to an unknown location. Numerous allied units in Westwood's Red Alert have teleporation and or invisibility modes based off of the Philadelphia Experiment. The experiment whether hoax or not has had a ripple effect in movies, games and novels. The aspect of teleportation and invisibility have also been explored in the military with bombers such as the B-2 being able to have radar invisibility, but not actual cloaking. Most likely, no one will ever know if this event occured or not, or if it is hoax or not, but to me it seems like popular science fiction.