hexakosioi hexekonta hex. - Rev 13:18
That is, 666. This is the way I was taught the numerology of it:
Six is man's number. (Man was created on the sixth day according to Genesis 1.) Seven is God's number. (God rested on the seventh day, Gen 2; plus a whole lot of "seven" references since then.) Because man is fallen and God is perfect, six and seven also become references to imperfection and perfection respectively. (But that's not important here.) Three is also a number related to God: it means the Trinity, for one.
Three sixes, "666", are said to represent a man (six) trying to be God (three). This goes with the context of the passage:
I saw another Beast rising out of the ground. It had two horns like a lamb but sounded like a dragon when it spoke. It was a puppet of the first Beast, made earth and everyone in it worship the first Beast, which had been healed of its deathblow.
This second Beast worked magical signs, dazzling people by making fire come down from Heaven. It used the magic it got from the Beast to dupe earth dwellers, getting them to make an image of the Beast that received the deathblow and lived. It was able to animate the image of the Beast so that it talked, and then arrange that anyone not worshiping the Beast would be killed. It forced all people, small and great, rich and poor, free and slave, to have a mark on the right hand or forehead. Without the mark of the name of the Beast or the number of its name, it was impossible to buy or sell anything.
Solve a riddle: Put your heads together and figure out the meaning of the number of the Beast. It's a human number: six hundred sixty-six.
Rev 13:11-18, the Message translation
The idea of 666 as meaning a man trying to be God is also visible in the "anti" in Antichrist, from Greek anti meaning "against" or "in place of".
Some, mostly Protestants, associate the first beast ("the" beast) with the papacy. For example, the pope's title, "Vicarius Christi", which means vicar of Christ, "vicar" from Latin vicarius meaning "substitute". Some point to fulfillments of prophecies in Revelation, and some add up one of the pope's archaic, unofficial titles: "Vicarius Filii Dei" (vicar of the Son of God) which, in Roman numerals adds up to the number of the beast, 666.