1 It is derived from the Bible
The number 666 was mentioned in the christian's bible in the book of revelation. The number was referring to the incompleteness of man, his sins and to the Devil who motivated man to sin.
2 The number 6 is incomplete
Gematria is a system of numerology in which every letter of the Hebrew alphabet is assigned a numerical value and special meaning. According to that system, the number 1 refers to God, the number 7 refers to the days of creation, while the number 6 is considered an incomplete number.
3 Humans were created on the 6th day
According to the Bible, Man was created on the 6th day. Because man chose to sin and disobey God, the number 6 was associated with sin and evil.
4 The number refers to the beast or Satan
According to the Book of Revelation in the Christian's Bible, 666 is the number associated with the beast. The beast, according to the Book of Revelation, is the Devil.
5 A reference to the evilness of money
According to a Quora thread, 666 could be referring to the big profit King Solomon made in one day. According to the thread, the number was considered evil because money can be evil. The thread quoted the verse 'You cannot serve both God and Money' from the Book of Matthew. (See Is Seven Really a Lucky Number?)
6 It might be referring to Emperor Nero
Using Gematria techniques, words can be derived from numbers. The number 666 could possibly be referring to Emperor Nero who prosecuted and killed Christians.
7 It referred to the Sun God
According to one theory, the number 666 was used by the Babylonians to refer to the most supreme God they believed in, which is the God associated with the Sun. Christians could thus have considered the number evil as it referred to a pagan God. (See Which are the 66 books of the Bible, in the same order as listed in the book?)