The Four Crowned Princes of Hell

Satan
Meanings & associations: Adversary, opposite, accuser, Lord of Fire, the Inferno
Element: Fire
Cardinal direction: South

"Invocation to Satan" from The Satanic Bible:

"In nomine dei nostri Satanas Luciferi excelsil.  In the name of Satan, the Ruler of the earth, the King of the world, I demand the forces of Darkness to bestow their Infernal power upon me!  Open wide the Gates of Hell and come forth from the abyss to greet me as your brother (sister) and friend!  Grant me the indulgences of which I speak!  I have taken thy name as a part of myself!  I live as the beasts of the field, rejoicing in the fleshly life!  I favor the just and curse the rotten!  By all the Gods of the Pit, I command these things of which I speak shall come to pass!  Come forth as answer to your names by manifesting my desires!"

Satan represents, to the Satanist, rebellion and opposition.  Satan represents justice and doubt.  The Satanist exists on the other side of the tracks.  Conformity for conformity's sake is seen as mindless and vapid, and the Satanist seeks to establish his own unique identity---the rebellious character of Satan is an ideal symbol for a person of this disposition.  Satan is the dark force in nature and represents brutal reality in all its blood-and-guts glory, rather than looking at the world through rose-tinted glasses and telling yourself the popular comforting lies about life and other people.

"Satan represents undefiled wisdom instead of spiritual pipe dreams."  "Satan represents man as just another animal, sometimes better but more often worse, than those that walk on all fours, who, because of his divine intellectual and spiritual development, has become the most vicious animal of all." ---Anton Szandor LaVey, The Satanic Bible

Satan was not originally perceived as one supernatural entity; rather, the word "Satan" was used to describe any adversary or opposer of the Hebrew God.



Lucifer
Meanings & associations: bringer of light, enlightenment, the air, the morning star
Element: Air
Cardinal direction: East

The Satanist seeks "undefiled wisdom."  He seeks to know the truth and embrace reality as fully as possible, because, as The Satanic Bible says, "Popular lies have ever been the most potent enemies of personal liberty.  There is only one way to deal with them: Cut them out, to the very core, just as cancers.  Exterminate them root and branch.  Annihilate them, or they will us." 
LaVey also said, "Satanism demands study, not worship."  The Satanist seeks knowledge and self-betterment in all the ways that he can.  Lucifer is an ideal symbol for this, representing enlightenment and the gaining of knowledge.  Biblically speaking, Lucifer was Satan's name prior to his "Fall."  He was said to be the angel of light and music; Luciferians have retained this symbolism and stressed these associations.  While Christ praises ignorance in the Bible, Lucifer advocates self-improvement and personal growth.
Lucifer is mentioned only once in the Hebrew Bible, and is most likely a reference to Babylon, and was not originally associated with Satan.  Later developments and fictional works such as Milton's Paradise Lost and Dante's The Inferno further established the association/equation of Satan and Lucifer.



Belial
Meanings & associations: without a master, baseness of the earth, independence
Element: Earth
Cardinal direction: North

"Belial is the carnal side of man, the lust, sex, pleasure and therefore the principal drives that make living worthwhile. People derive all the principal emotions of the higher ego from Belial: Pride comes from self control and suppression of the Belial, strength, pleasure and independence come from embracing it. Belial is the Master of the Earth, the force that holds Humankind by its balls, any security or stability are results of lessons learnt from dealing with this Crown Prince.
Belial is the champion of simply being human, for the flesh, the material and the carnal. In essence, a reverence for Belial affirms how "good" the flesh/humanity is. Unrestrained by law or morality; lawless; immoral; dissolute; lewd; lascivious, Unrestrained; uncurbed; uncontrolled; unruly; riotous; ungovernable; wanton; profligate; dissolute; lax; loose; sensual; impure; unchaste; lascivious; immoral, dissolute indulgence in sensual pleasure." ---Vexen Crabtree

"Belial means "without a master", and symbolizes true independence, self-sufficiency, and personal accomplishment. Belial represents the earth element." ---Anton Szandor LaVey, The Satanic Bible

"But for corruption thou hast made Belial, an angel of hostility. All his dominions are in darkness, and his purpose is to bring about wickedness and guilt. All the spirits that are associated with him are but angels of destruction." ---Dead Sea Scrolls, "The War of the Sons of Light Against the Sons of Darkness"



Leviathan
Meanings & associations: the serpent of the deep, the sea
Element: Water
Cardinal direction: West

Leviathan was a sea serpent, mentioned in the Biblical Book of Genesis.  In Satanism, he represents terror, power, desire, secrecy, sexual desire, the unknown, and inner emotion.  The Hebrew Bible has the following to say about him:

¹⁰ No-one is fierce enough to rouse him. Who then is able to stand against me?

¹¹ Who has a claim against me that I must pay? Everything under heaven belongs to me.

¹² I will not fail to speak of his limbs, his strength and his graceful form.

¹³ Who can strip off his outer coat? Who would approach him with a bridle?

¹⁴ Who dares open the doors of his mouth, ringed about with his fearsome teeth?

¹⁵ His back has rows of shields tightly sealed together;

¹⁶ Each is so close to the next that no air can pass between.

¹⁷ They are joined fast to one another; they cling together and cannot be parted.

¹⁸ His snorting throws out flashes of light; his eyes are like the rays of dawn.

¹⁹ Firebrands stream from his mouth; sparks of fire shoot out.

²⁰ Smoke pours from his nostrils as from a boiling pot over a fire of reeds.

²¹ His breath sets coals ablaze, and flames dart from his mouth.

²² Strength resides in his neck; dismay goes before him.

²³ The folds of his flesh are tightly joined; they are firm and immovable.

²⁴ His chest is hard as rock, hard as a lower millstone.

²⁵ When he rises up, the mighty are terrified; they retreat before his thrashing.

²⁶ The sword that reaches him has no effect, nor does the spear or the dart or the javelin.

²⁷ Iron he treats like straw and bronze like rotten wood.

²⁸ Arrows do not make him flee, sling stones are like chaff to him.

²⁹ A club seems to him but a piece of straw, he laughs at the rattling of the lance.

³⁰ His undersides are jagged potsherds, leaving a trail in the mud like a threshing-sledge.

³¹ He makes the depths churn like a boiling cauldron and stirs up the sea like a pot of ointment.

³² Behind him he leaves a glistening wake; one would think the deep had white hair.

³³ Nothing on earth is his equal—a creature without fear.

³⁴ He looks down on all that are haughty; he is king over all that are proud.

----Job 41: 10-34