Answer: Strictly speaking, the word "incest" is not found anywhere in the King James translation. The overall basic principle, however, that defines this act is located in Leviticus 18:6. It states, "None of you shall approach to any who are near of kin to him to uncover their nakedness (a euphemism for having sex)." The rest of the chapter gives us specific examples of how the Bible defines incest and other forbidden kinds of sexual relationships (whether it leads to marriage or not).
Verses 7 through 15 of Leviticus 18 specifically forbids sex (incest) between a man and their mother (or stepmother) or father, sister (or half-sister), daughter-in-law, granddaughter, aunt (by blood or marriage), or sister-in-law. Verse 16 forbids intercourse with a brother's wife.
There is an Old Testament exception, however, to all the above. It occurs when a man's brother dies without a son. In such circumstances, the Bible states a man was duty-bound to marry and produce children with his brother's widow (Deuteronomy 25:5 - 10). In this special case sexual perversion was not imputed.
Verse 17 of Leviticus 18 declares it is incest for a man to have a sexual relationship with a woman and her daughter (a daughter from a previous relationship, a stepdaughter), either at the same time or with the daughter if the woman dies. It also forbids such a relationship with the daughter of either the woman's daughter or son (step-granddaughter). Verse 18 prohibits taking a woman and her sister as rival wives.
The verses quoted above apply to both men and women, and covers even relationships assumed but not directly named in Leviticus (such as sex between a parent and their children).
Can you marry a cousin?
Interestingly, Scripture does not specifically forbid the marriage of cousins. However, because of 6,000 years of genetic mutations, the chance of passing on genetic defects to the children of such a union might make such a relationship unwise (it might also be illegal where you live).
Scripture contains examples of incest that were allowed (tolerated) before the giving of the law. For example, God allowed Adam and Eve's children, such as Cain and Seth, to marry their sisters out of necessity. Abraham was allowed to marry his step (half) sister Sarah (Genesis 20:12). Lot had intercourse with his daughters after being made drunk by them (Genesis 19:31 - 32).
New Testament teaching
By the time of the New Testament, however, incest was strictly forbidden. John the Baptist condemned the marriage of Herod Antipas to his brother Philip's wife (Mark 6:17), an act that was clearly a sin (Leviticus 18:16, 20:21). The apostle Paul had to grudgingly deal with a case of incest occurring in the church at Corinth. The church had a member who was having sex with his stepmother while his father was still alive (1Corinthians 5)!
God's law enumerated differing penalties for those found committing incest. They included exclusion from the covenant people (Leviticus 18:29, 20:17 - 18), childlessness (Leviticus 20:20 - 21) or even death (Leviticus 20:11 - 12, 14). The reason why he forbids this behavior (other than he says so) is that those who practice it defile and destroy themselves through it (Leviticus 18:24 - 25, 27).
In conclusion, the Bible not only does not approve of incest, it also forbids we have sex with anyone (even if they are not closely related to us) outside the union of marriage. In fact, according to Jesus, even lusting after someone (before any physical act can take place) is considered a sin (Matthew 5:28)!