There are 3 astrological points with the name Lilith: asteroid Lilith, Black Moon Lilith and Dark Moon Lilith. They are all called Lilith because the way you interpret them in a chart is related to the Lilith story.
Probably the earliest version of Lilith is from Mesopotamia in the 13 -15th centuries BC where she was the Goddess of Death in the form of an owl. She was called Lilith because it means "screech owl". The version of Lilith we know today is as the first wife of Adam, before Eve, although she is not mentioned in the Bible or in ancient Hebrew religious writings. This version seems to come from a medieval writing that purports to be part of Hebrew writings but is not. The writer probably took the older legend and Christianized it by making it part of a Bible story. Lilith became the woman from hell, the anti-female.
There are many versions and interpretations of her story. As the first wife of Adam she refused to be subordinate to him. She refused to lie beneath him in sexual relations. She wanted to lie side by side or on top. When he tried to force her she used her powers to fly away. This part of her story has interested many women today . They like to think of Lilith as a feminist heroine, saying she was courageous and asserting her equality. However, in the original story she was more a symbol of the psychologically damaged. If you interpret the Liliths in a chart as heroic or admirable it will not be accurate. The Liliths in your chart operate as places where you are having trouble, where you don't fit in with the norm.
After Adam refused to have her as his wife, she ran off and mated with many demons, which was strictly against God's will. God sent 3 angels to her to get her to stop but she didn't obey so He started killing her children at the rate of 100 per day. She went mad with grief and started killing other people's children. Painful issues surrounding children, birth, unsuitable sexual partners and parts of your life where you just can't function like others do are Lilith issues.
The glyph for asteroid Lilith is a straight horizontal line with 4 small vertical lines sticking up from it , sort of like an upturned rake head, and on the right below the horizontal line is a very small mark like a capital L or V. Some say it is an upturned hand which could be either rejecting or expecting something different. It takes about 5 years for Lilith to travel through all 12 signs, since it does retrograde a lot. It orbits the Sun in the asteroid belt between Mars and Jupiter. The asteroid Lilith has to do with all aspects of difficult sexual relationships.
Black Moon Lilith
Black Moon Lilith is the astrologer's term for the lunar apogee. Since the Moon has an elliptical (oval) orbit and a wobbly one at that, the distance the Moon is from the Earth varies. The lunar apogee is the mathematical point in the sky where the Moon is farthest from the Earth. The reason this point is called Lilith is because in the Lilith story she was acting out the difficult (distant) part of the Moon, in murderous rage, vindictiveness, and insanity. The glyph for Black Moon Lilith is a black crescent pointing east, similar to a 3rd quarter Moon, but with a cross underneath.
Black Moon Lilith can be calculated as mean or true, similar to the way the Moon's nodes are calculated as mean or true. But in the case of the Black Moon there can be as much as 30 degrees difference between the mean and true so it is possible for them to be in different signs or houses. Deciding which to use is easier if you look at charts of people you know where the mean and true are in different signs and houses. The true is the more accurate placement but an ephemeris for it has only been available for about 10 years so not as much work has been done using it. The ephemeris for the mean has been available for about 100 years and astrologers in Europe have been using it almost that long. That's why you see the mean used more in articles and books.
The mean moves about 39 degrees per year or about 3 deg 8 min per month and does not retrograde. The true Black Moon retrogrades a lot and can move anywhere from 9 deg to well over 70 deg per year. It also can move up to 4 deg per day or stay at the same degree for several days so it is important to get an accurate placement for the true. An ephemeris in a book is fine for mean but not for the true. You can get the true in Win*Star software, Halloran software, or Riyal freeware or you can use the Swiss Ephemeris online to get both true and mean. Directions for using the Swiss Ephemeris to find Black Moon Lilith are at http://groups.yahoo.com/group/3Liliths Click on 3 Liliths Info, then scroll down to the Black Moon section. It takes about 9 years for the mean or true to pass through all 12 signs.
Dark Moon Lilith
Unlike Black Moon Lilith, Dark Moon Lilith is a real body in the sky, a second Moon to the Earth that doesn't reflect light, hence the name Dark Moon. The glyph for Dark Moon Lilith is a zero with a forward slash running diagonally through it. There is some controversy over whether it is really there since it can only be seen as a spot moving across the Sun. However, there have been sightings since the 1600's as detailed in Delphine Jay's book, Interpreting Lilith, pg. 1-6. Dr. Georges Waltemath, an astronomer, documented it in 1898 and came up with calculations which gave it an orbit of 119 days around the Earth.
The astrologer Sepharial used the Waltemath calculations to come up with an ephemeris but he made errors which caused a difference of anywhere from about 2 to over 40 degrees. Delphine Jay, Ivy Goldstein-Jacobson, Mae Ludlum-Wilson and others copied Sepharial's calculations. The original Waltemath calculations can be found in Solar Fire software or Riyal freeware. Many astrologers use the Sepharial calculations with success, but the Waltemath calculations are more accurate. Dark Moon Lilith can be interpreted similarly to the other Liliths. Delphine Jay says Dark Moon Lilith is a place where you must operate impersonally because operating personally there will never be successful.
Further information on all 3 Liliths can be found at http://groups.yahoo.com/group/3Liliths/