Who/what is Lilith?
Lilith in Mesopotamia
First Lilith (Lilitu) is of Mesopotamian origin. She is a female "demon"/"succubus" (Consider I use here modern terms, Sumerian or Akkadian got no clue what those was). We find her in diverse stories most notable one is when she is coming to haunt Inanna's tree in Inanna and the Hullupu tree (We don't know what an Huluppu tree is).
The years passed; five years, and then ten years.
The tree grew thick, But its bark did not split.
Then the serpent who could not be charmed Made it's nest in the roots of the huluppu-tree.
The Anzu-bird set its young in the branches of the tree.
And the dark maid Lilith built her home in the trunk.
Note you find various versions and translations, but the idea is always the same. Lilith is vanquished by Gilgamesh there.
In the original text, the Huluppu tree is named halub[GISH-HA-LU-UB] or hulub[GISH-HU-LU-UB], Haluppu or Huluppu in Akkadian:
ubia gishdisham, halubdisham gishdisham
On thay day, a tree, a Huluppu tree, a tree
Lilith is ki-sikil-li-la-ke (female[kisikil] Lilake[demon]). Lilu in Akkadian in "a demon", its female form is lilitu:
shabbabia kisikillilake, e imanibdu
In it, Lilith had built a house
She is related with the death of newborn children and such bad things.
Lilith in the Holy Bible
First, there is not even one track of Lilith, per se, in the Holy Bible. The one of the Christians. That said the Hebraic version will clarify what the Bible says:
26 Then God said, “Let us make mankind in our image, in our likeness, so that they may rule over the fish in the sea and the birds in the sky, over the livestock and all the wild animals,[a] and over all the creatures that move along the ground.”
27 So God created mankind in his own image, in the image of God he created them; male and female he created them.
It is a well-known thing that here men and women are created from clay at the same time. BUT here the woman is created AFTER from a man's rib:
But for Adam[f] no suitable helper was found. 21 So the Lord God caused the man to fall into a deep sleep; and while he was sleeping, he took one of the man’s ribs[g] and then closed up the place with flesh. 22 Then the Lord God made a woman from the rib[h] he had taken out of the man, and he brought her to the man.
That is the version most people know. But the Bible is still quite unclear if only Eve was a part of a rib or all the females.
Lilith for the Hebrews
For the Hebraic books, Adam got a first wife, precisely Lilith (sometimes named Adamah), modeled out of clay. And they divorced. There are several traditions according to the reason of their separation. The most renowned (but well truly a joke) is that she wanted to be equal with Adam during sex. Which was not his taste, hence she was expelled from Paradise. This is from the Alpha Beta. (This is THE tradition you find everywhere.) Just here, The Alpha Beta (or Alphabet) 78 (Note this work is renown for being a gigantic satire):
When God created the first man Adam alone, God said, “It is not good for man to be alone.” [So] God created a woman for him, from the earth like him, and called her Lilith. They [Adam and Lilith] promptly began to argue with each other: She said, “I will not lie below,” and he said, “I will not lie below, but above, since you are fit for being below and I for being above.” She said to him, “The two of us are equal, since we are both from the earth.” And they would not listen to each other. Since Lilith saw [how it was], she uttered God's ineffable name and flew away into the air. Adam stood in prayer before his Maker and said, “Master of the Universe, the woman you gave me fled from me!”
Another from Zohar says that she was not Adam's first wife or just a concubine and she became over jealous when Eve was created.
After the primeval light was hidden, a husk was created for the brain, and that husk spread out and brought forth another husk which was Lilith. And when she emerged, she went up and went down towards the little faces, and wanted to attach herself to them and be shaped after them, and did not want to depart from them. But the Holy One, blessed be He, removed her from there and placed her down below. When He created Adam, in order to perfect this world, as soon as Lilith saw Eve affixed to the side of Adam, and saw in them the beauty of the Above, and saw their perfect image, she flew off from there and wanted, as before to attach herself to the little faces.
The tradition also usually makes her the snake that charmed Eve coiled around the tree of Life (cf. Lilith in Sumer's myth stealing Inanna's tree).
As much as I dislike giving personal opinion on things, here it is difficult to a least resist commenting a little bit.
The alphabet is presenting a couple where the woman wants to do some "cowgirl" and the man refuse! And for that Lilith is giving pretty bad name to God and flee. You clearly see why the Christians do not care about that book! And it is obvious why the book is considered nonserious.
The Zohar story is slightly different. Here Lilith appears as a concubine of some sort and without being consulted on the matter, God creates a new woman. Let's face it, it is difficult to not understand Lilith's being seriously pissed off. Which also push me to notice the Zohar should be taken with high caution.
Lilith in Demonology
Demonology also knows Lilith as one of the 4 queens of Hell with Na'amah, Agrat Ba'at Malhat, and Eischeth Zemium. Consider her as a wife/Consort of the ruler of Hell.
The ruler can be Samaël, Asmodeus, Beelzebub, etc. And sometimes you see Lilith the Old as Samaël's devoted wife when you find a Lilith the Young in love with Asmodeus.
Lilith for the feminists
The feminists love Lilith. It is as I mentioned vastly renown that Lilith was Adam first wife carved from clay, as him, and wanted to be equal. Lilith is so for this behavior very famous among the feminists' aficionados. Lilith becoming an icon of the woman's liberty. Now it seems the feminists did not check the seriousness of their sources (or simply not checked them or worst) just because basing the men/women equality on such a book as the alphabet is... funny.