"The small island of Pátmos in the Dodecanese is reputed to be where St John the Theologian wrote both his Gospel and the Apocalypse. A monastery dedicated to the ‘beloved disciple’ was founded there in the late 10th century and it has been a place of pilgrimage and Greek Orthodox learning ever since. The fine monastic complex dominates the island. The old settlement of Chorá [the capital 'city'], associated with it, contains many religious and secular buildings." Source: UNESCO World Heritage List.
"Midway along the road that winds steeply up from Skála [the only commercial port on the island] to Chorá is the Cave of the Apocalypse ... where according to tradition St John dictated the Book of Revelation and his Gospel to his disciple Prochoros. This holy place attracted a number of small churches, chapels, and monastic cells, creating an interesting architectural ensemble."
According to tradition, John had been exiled to Patmos "as a result of anti-Christian persecution under the Roman emperor Domitian, but it's also possible that Patmos was one stop on an established Christian pilgrimage route. John's identify is up in the air:
"Traditionally, the writer of Revelation is widely considered to be John the Apostle, who is also seen as author of the Gospel of John (1). However, others identify the author as John the Elder and many modern scholars believe it was written by an otherwise unknown author, to whom they have given the name John of Patmos." It's also possible that Revelation was written by a completely unknown apocalyptic prophet who passed off the book as the work of one of these other characters in order to make it seem more authentic.
(1) It's unlikely that John the Apostle (assuming he actually existed) wrote the Book of Revelation since the emperor Domitian didn't come to power until 81 AD. The dates here are kind of shaky and as far as I can tell there isn't much evidence that Domitian actually persecuted the Christians, but some historians believe that he launched a persecution against Jews and Christians during the latter part of his reign (89-96 AD). Though the Apostle John is thought to have lived from 6-100 AD, if these dates are correct, he would've been somewhere between 83 and 90 years old when Revelation was written. It's still possible, however, since according to the traditional story he dictated the book to Prochorus.