There is no one by the name of Lilith mentioned in the Bible. There is a myth in Gnostic creation mythology of Adam’s wife being of that name. I will not try to explain Gnostic mythology but rather uncover the use of the word lilith in the Bible. The only time that the word appears in the Bible is at Isaiah 34:14, where the Hebrew word lilith is normally translated as “nightjar”. Which is a type of owl found in the middle east. Here it is Isaih 34:14 " And haunters of waterless regions must meet up with howling animals, and even the goat-shaped demon will call to its companion. Yes, there the nightjar will certainly take its ease and find for itself a resting-place". Some scholars believe the term li·lith´, used at Isaiah 34:14 as among the creatures haunting Edom’s ruins, applies to some type of owl. The name is said to be used today “for Strix, the tawny owl.” (The Interpreter’s Dictionary of the Bible, edited by G. A. Buttrick, 1962, Vol. 2, p. 252) NIGHTJAR [Heb., li·lith´]. A creature appearing in the description of Edom’s utter desolation and of the things inhabiting its ruins. (Isa 34:14) The Hebrew word has been variously translated as “screech owl” (KJ), “night-monster” (AS), “nightjar” (NE, NW), and “night hag” (RS), while The Jerusalem Bible prefers simply to transliterate the name as “Lilith.” Many scholars endeavor to show that the Hebrew term is a loanword from ancient Sumerian and Akkadian and that it derives from the name of a mythological female demon of the air (Lilitu). Professor G. R. Driver, however, considers the Hebrew word (li·lith´) to derive from a root word denoting “every kind of twisting motion or twisted object,” even as the Hebrew word la´yil (or lai´lah), meaning “night,” suggests a “wrapping itself round or enfolding the earth.” Such derivation of li·lith´, he suggests, may likely point to the nightjar as both a nocturnal feeding bird and one noted for its rapid twisting and turning flight as it pursues moths, beetles, and other night-flying insects. As quoted by Driver, Tristram, the naturalist, described the nightjar as “becoming very active towards dusk, when they hawk about at great speed and with intricate turnings after their food.” Palestine Exploration Quarterly, London, 1959, pp. 55, 56. The nightjar is almost 30 cm (12 in.) in length with a wingspan of 50 cm (20 in.) or more; its plumage resembles the owl’s, being soft and delicately mottled with gray and brown. The soft wing feathers allow for noiseless flight. Its large mouth is evidently the reason for its also being called the goatsucker, an ancient legend holding that the bird sucked the milk of goats. The idea that Adam had a companion named Lilith is not Biblical.

Many different Liliths exist. Since you refer to the creation story, I will address that one. All references to "God" are in relation to this myth. In this myth "God" means Yahweh/Jehova. Jewish and Gnostic oral tradition holds that Lilith was the first wife of Adam. She was the one first created "male and female He created them", and the second story about the rib is about Eve. Lilith, however, refused to lie beneath Adam during sex (i.e. she refused to submit to Adam). Differing versions of the myth have her aquiring her wings in different ways, but she flew away from Adam. The myth gives her the power of knowing God's name, and - therefore - some measure of power over Him, and the ability to defy Him. Lilith is portrayed as a beautiful woman with wings and talons for feet. She was feared to prey on infants. Thus, a talisman was hung around their necks or hung above their crib. She is also credited with giving birth to many demons. She herself is quite often characterized as a demon. http://en.wikipedia.org/wiki/Lilith

The story AntigoneRising gives is correct as far as what I've been taught. There are many alternate stories about the creation that were common in midevil times but have fallen out of favor nowadays. Adam actually had three wives according to some forms of apocrypha. The first wife was never given a name - Adam watched her be created by God, but was so disturbed by the sight that he refused to associate with her. God uncreated the first wife, and created Lillith while Adam was asleep. Lillith was a full creation (from scratch) and had a will of her own, and refused to be dominated by Adam sexually or otherwise. This didn't work out either. Instead of being uncreated Lillith was allowed to flee the garden, and has since figured into other myths, sometimes with decendents. She frequently features as a succubus, who preys upon men as they sleep and draws out their life force, possibly through sex. Then God created Eve, while Adam slept, drawing her from the body of Adam (and therefore, by implication, making her subordinate to him). Adam was pleased with this wife. But as we know, this didn't work out so well in the end, as this version of Woman was not perfect either, and tempted Adam into sin. Those darn womens, they're just no good. ;)

The Bible does not tell us the whole story. Lilith was a powerful angel second powerful to Lucifer. Lucifer's love for Lilith made God jealous, plus the thought of Lucifer and Lilith using their joint power against God worried Him as their joint powers would have been the mightiest in the universe. So when God created man (Adam) he decided to bring Lilith down to earth as Eve. The spirit of Lilith in Eve is what made her take the bite of the forbidden 'fruit'in a rebel rage against God for making her human. Lucifer constantly reminded Eve who she really was and Eve couldn't help being drawn to the angel of light, Cain was the result of their union. Adam was drawn into the 'sin' by accepting Cain as his own son and Eve as his wife even though he was aware of what had happened. Hence the fall of Lucifer and Man.