Lilith is a mythological character purported to have been Adam's first wife. The Bible contains no such account nor even hints at such a possibility. According to the legend, Lilith was headstrong and independent, and didn't want to submit to Adam, so she divorced him. How the Lilith legend came to be is a circuitous tale on its own.
Who was Lillith / Lilith? Did Adam have another wife before Eve?
The legend of Lilith originated in the last chapter of the Epic of Gilgamesh—a chapter which was probably not original to the rest of the text. In the story, the goddess Inanna finds a tree in a river and plants it in her garden. She cares for it for ten years, but finds that it's been infested with "the serpent who could not be charmed," "the Anzu-bird," and "the dark maid Lilith." Inanna cannot get rid of the squatters, so she asks her brother Gilgamesh. He strikes the serpent, leading the Anzu-bird to flee with its young and Lilith to smash her home and escape to "the wild, uninhabited places." Gilgamesh chops up the tree and makes a throne and a bed for Inanna.
The Aleppo National Museum is in possession of an amulet with the engraving of a sphinx and a she-wolf that includes the words "O, Demoness-that-flies in a dark chamber, Get on your way at once, O Lili!" The amulet is thought to be Syrian, from the 6th or 7th century BC, but it's also possible it's a forgery from the 1930s.
It's possible that the Bible references Lilith as a pagan character. Isaiah 34:14 reads, "And wild animals shall meet with hyenas; the wild goat shall cry to his fellow; indeed, there the night bird settles and finds for herself a resting place." The night bird (some translations say "screech owl") is the Hebrew Liyliyth. It is derived from layil, which means "night." "Lilith," literally, means "night maid," so it's unclear if the verse refers to the Sumerian goddess or if it's a poetic way to describe a female nocturnal bird.
Some argue for Lilith's existence by pointing to the seeming conflict between Genesis 1 and 2. In Genesis 1:27, God creates male and female. But Genesis 2:18–25 tells the story of the creation of Eve. In truth, Genesis 1 is a synopsis of the six days of creation while Genesis 2 gives more detail into the sixth day when God created Adam and Eve. But many people misinterpret the timeline and think the chapters are straight chronology. Genesis 1:27, they say, refers to Lilith.
Extra-biblical writings of Judaism hold to this account. The midrash Genesis Rabba (AD 300—500) infers that Adam had a first wife. The Babylonian Talmud says that Lilith has wings, that she can cause birth defects, that she is a succubus, and that she used the nocturnal emissions of sleeping men to conceive demon babies. The first text that clearly connects Lilith as Adam's first wife is The Alphabet of Ben Sira. In this text, Lilith is said to have left Adam when he demanded she be submissive in sex. When Adam asked angels to bring her back, she said she wouldn't. The angels told her they would kill her demon children, so she responded that she would in turn kill the babies of the descendants of Adam.
Further legend says that she is responsible for diphtheria, stillborn children, and babies who die of SIDS. It was a short leap to go from Lilith as "night maid" to "night hag," and blame her for sleep paralysis. Some, including Michelangelo, associate her with the serpent that tempted Eve. In this incarnation, she is the wife of Satan and provides the body so that he can be the voice that talks to Eve.
More recently, feminists and New Agers have claimed Lilith as a role model. They praise her independence and sexual freedom and use her as an example when refusing to submit to their husbands. She has leant her name to "Lilith Fair," a touring concert of female singers and female-led bands, and Lilith Magazine, a Jewish feminist magazine.
The only verified part of all this is that Lilith was a character in ancient Sumerian/Akkadian folk lore. Any "evidence" found in the Bible is easily dismissed. The bulk of literature defining her role in history comes from Kabbalah—a Jewish-based cult. In short, Lilith was a figure of ancient mythology who has since been used to represent death to innocents, sexual predation on men, and feminist independence. She was never real, and she certainly was never married to Adam.
Is the Adam and Eve story true? Is it literal?
Do Genesis 1 and 2 record two different creation accounts?
What is the Epic of Gilgamesh? What relation does it have with the biblical Flood?
Why did God create humanity?
What is the importance of biblical creationism?
Truth about Creation