Biography
The Villain Song is an over-the-top, gloating cackle about their Evil Plan, philosophy of life, or why they do what they do (their leitmotive?). Said philosophy is usually that the world is there for the taking, or that the world has wronged them and they will take revenge, or that they just like hurting people and nobody is going to stop them. The villain usually gets lots of eye candy to go with their song — live performers will dance up a storm, while animated villains will get pyrotechnics all around them. If they don't get huge visuals, it better damn well be because the song itself steals the show, and trying to distract from that with a lot of flashy visuals is just asking for a chandelier to fall on someone. This song is usually delivered in a minor key and typically features dramatic evidence of the villain's evil nature, such as playful abuse of loyal minions, taunting of prisoners, or random destruction of incidental locations of the villain's secret lair, usually for the purpose of abusing minions or taunting prisoners.
The nuttier the villain is, the better the song is. The Barnum (see "Master of the House" from Les Misérables) and the Straw Nihilist (see "Epiphany" from Sweeney Todd: The Demon Barber of Fleet Street) get particularly good ones.
This song is almost always a measure of how important the villain is. If the villain has one, they'll get equal billing with the hero, and will probably steal the show. If they don't (e.g. Dr. Carrasco in Man of La Mancha, Charlie Cowell in The Music Man), they're not very important to the plot, and nobody qualifies as the Big Bad. Villains can get other songs where they pretend to be nice, but if they don't get at least one song to strut their villainy, they're (usually) not important (the big exception being villains who successfully offer a Deal with the Devil: There, the hidden malice and seductiveness is usually the point of the song anyway).
Occasionally followed up by an Evil Laugh and/or a Fade Around the Eyes immediately, and a Dark Reprise later on.
Depending on genre, it can overlap with Creepy Jazz Music, Rotten Rock & Roll, Freaky Electronic Music, or Sinister Tango Music. It is by no means limited to these genres, though.
Compare Villain Love Song (where the villain sings about their admiration towards an object of affection, which can overlap), "The Villain Sucks" Song (which is sung about the villain by someone else), and Rock Me, Asmodeus! (the devil directly involved in rock & roll).
Contrast No Song for the Wicked. Now has its own Wiki.
Disney has had so many examples that they get their own page. All examples from media by Disney go there, while all the other pages are strictly for non-Disney examples from the media they cover.
Please note that just because a song is sung by a villain, doesn't necessarily make it a Villain Song. The Phantom of the Opera may have a showstopper in the form of "The Music of the Night", but he doesn't use it to gloat over his villainy. Villain Songs don't have to be personally sung by the villain either (see the title song from Spaceballs), though exceptions are fairly rare outside examples of "The Villain Sucks" Song.