The significance of the number 666 comes from Revelation 13:18:
This calls for wisdom. If anyone has insight, let him calculate the number of the beast, for it is man's number. His number is 666.
Most scholars believe that this number is a cryptogram for Nero Caesar. This hasn't stopped many people being accused of being "the beast" -- various popes, Martin Luther, Napoleon, Saddam Hussein, Prince Charles, and Ronald Reagan. (Nancy Reagan tried to put us off the trail by changing the address of their home in Bel-Air from 666 to 668.)
But, like the number 13, because it has an artificial significance, you notice when things pertain to it. For example, it is part of every UPC barcode on groceries -- a coincidence, according to the code's inventor, although I suspect "joke" instead. With biblical coding, 666 can also be the number for the WWW of the World Wide Web. Or not. For several years DC Comics headquarters were at 666 Fifth Avenue.
A lot of people who have a pop-culture understanding of the book of Revelation, rather than an understanding based on, you know, actually reading the text, don't know that there are at least three big beasty things in Revelation: a great red dragon with seven heads and ten horns, and seven diadems on his heads, which is thrown down to earth; a beast that rises up out of the sea and also has ten horns and seven heads, with ten diadems on its horns, and blasphemous names on its heads, like a leopard, with feet like a bear's, and a mouth like a lion's mouth, and is given authority by the dragon; and finally a beast that rose out of the earth, with two horns like a lamb, that spoke like a dragon and exercised the authority of the first beast, and made the earth and its inhabitants worship the first beast. It is this third creature which is identified with the number.
Perhaps, most significantly -- if you count from 1 to 1,000, you will, at some point, hit the number 666!