A pentacle is a 5-pointed star. It can be positioned so that the points point in various directions. A presentation with one upwards-pointing spoke is the 'right way' up, pointing upwards. One with two downwards-pointing spokes is an "inverted" or "upturned" pentacle. It is by far the former, point-up pentacles, that have been used in history, and I will refer to them simply as "a pentacle".
The pentacle is an ancient symbol, with prehistorical origins. According to Christian lore, it was used by the Biblical figure Solomon and in a Christian and Gnostic sense is said to represent the 'victory of spirit over matter'1. There are many interpretations of its meaning and significance, but what must be admitted is that it has been used by many people, at many times, to mean a variety of different things. The ultimately respectable ancient historian Prof. Hutton writes a little on its past:
“Five-pointed stars are found in ancient Egyptian, Greek and Roman art, and also in that of the Christian early Middle Ages, but there seems to have been no single tradition concerning their meaning and use, and in many contexts they seem simply to have been decorative. The enduring use of the symbol in the Western magical tradition is a product of the twelfth-century Renaissance, which put the pentagram at the center of its interplay between scripture, divine harmony, and mathematics. In that century both Honorius of Autun and Hildegard or Bingen asserted that the human body is constructed upon the basis of the number five, having five senses, five members, and five figures. This made the five-pointed star the symbol of the microcosmos, the earthly reflection of the divine plan and the divine image.”
"The Triumph of the Moon: A History of Modern Pagan Witchcraft" by Ronald Hutton (1999)2
It was used heavily by Christians throughout much of history, many times to represent Christ himself, and was used in such a way in the 12th to 16th centuries, also used as "an especially potent sign for the repulsion of evil"2. But it wasn't destined to always be a white light symbol. Christian paranoia and superstition about symbols led them, in history, to a multitude of purges and puritanical bans of whole swathes of religious symbolism. The pentacle began to fall foul of Christian censors, and eventually came to be seen by many of them as a non-Christian symbol, despite its Christian history. One author supplied a piece of invented theory that took this to its final step: That the pentacle could be used to invoke, rather than repel, demons. The head-case in question was Eliphas Levi, in 1854:
“[Eliphas Levi] restated another old tradition, that the five-pointed star controlled demons, and then extended it to assert that it specifically summoned and directed the demons of the four elements, from respective quarters of the consecrated circle. In 1854 he apparently took this to its logical conclusion; hitherto the symbol had been used only to ward off or banish spirits, but in that year Levi seems to have invented the invoking pentagram. He recommended the drawing of the sign in the air according to a particular pattern, above each quarter, to call up the elementals, and the drawing of it in the opposite direction to banish them at the end of the ritual.”
"The Triumph of the Moon: A History of Modern Pagan Witchcraft" by Ronald Hutton (1999)3
A pentacle with a single downwards-pointing spoke is an "inverted" or "upturned" pentacle. The inverted pentacle represents the carnal and 'ungodly' side of mankind rather than the spiritual. Various authors have concurred with this theme, although here our main concern is with the doctrine of the Church of Satan.
In the 19th century Eliphas Levi taught "that with two points down it was the symbol of God, with one point down, it was that of the Devil"4.
According to Akron, the inverted pentacle means 'victory of matter over spirit'1.
Aleister Crowley used it similarly "to signify the descent of spirit into matter"5.
Anton LaVey, the founder of the Church of Satan, describes the two anterior points as 'an attack on heaven'. It is an attack on the concept that Human Beings are toys of any god; a creation of any being or that there exists any being we should worship just because it exists; the Satanist reserves this privilege for himself only. LaVey also describes the posterior & two lateral points as a "denial of the trinity", referring to the Christian concept of the Trinity, codified in the 4th century. I prefer not to refer specifically to Christian doctrines and sometimes summarize the two lateral points as representing "two feet on the Earth"; i.e.; Belial. The attack on heaven & the right-hand-path is representative of Lucifer.
The inverted pentacle is a materialistic, human-centered symbol that like Satan represents a rejection of spiritual pipe-dreams such as heaven, and an embrace of the carnal and real elements of life.
The Sigil of the Baphomet is an inverted pentacle form, with two points pointing upwards, one downwards, and one to each side. It is the official symbol of the Church of Satan. Inside the pentacle is the image of a goat, its horns filling the upper two points, its ears filling the two lateral ones, and its mouth filling the lower one. This is Satan. The goat and pentacle theme is very old, and has its origins in alchemy. The pentacle is surrounded by two circles the innermost touching the points of the pentacle. In between the two circles are 5 Hebrew characters, spelling LVTHN anticlockwise: Leviathan.
A modern interpretation is that it represents fertility, success and power.
“The symbol of Baphomet was used by the Knights Templar to represent Satan. Through the ages this symbol has been called by many different names. Among these are: The Goat of Mendes, The Goat of a Thousand Young, The Black Goat, The Judas Goat, and perhaps most appropriately, The Scapegoat.”
“Baphomet. The term may have come from two Greek words, baphe and metis, meaning "absorption of knowledge." It has also been called the Black Goat, Devil's Goat, Goat Head, Goat of Mendes, and Judas Goat.”
The history of the Baphomet's usage by the Church of Satan is very clearly laid out with as much information as you could possibly want from a very good source ... the Church of Satan Official Website, as linked at the top and bottom of this page.