"Vanity" is another way of saying "pride". Of course that would be the Devil's favorite sin, being the reason he Fell and all.
Though it's highlighted a few times during the film, many people missed the Numerology. Kevin Lomax spent 5 years in a Florida court room. While there, he wins 64 straight cases. He goes to New York to continue his career. While there, he wins his 65th case, which was a health code case. And then he wins his 66th case which was about the business mogul committing multiple homicide. It's then that Satan reveals himself and tries to convince Kevin to mate with his half-sister and produce the Anti Christ. Why? Because afterwards, Kevin would be entering his 6th year as a lawyer with 66 straight cases under his belt - meaning 666. The fact Milton tells Kevin that he was meant to lose the homicide case in favor of saving his wife, makes this even more apparent. Once Kevin choose to win not matter what, he sealed his fate.
Kevin's mother mentions that his father (John Milton) knew 'the bible, every word'. As the old saying goes, even the devil can quote scripture.
When Satan gloats to Kevin about having sex with Mary-Anne, he says that "on a scale of one to ten, ten being the most depraved act of sexual theatre know to man, one being your average Friday night run-through at the Lomaxes' household, I'd say, not to be immodest, Mary Ann and I got it on at about... seven." This after Milton raped her, left her with gashes all over her body, and left her so traumatized that she later committed suicide. Now imagine what a ten out of ten must entail.