The image everyone has in mind was drawn by Eliphas Levi for his book on high magic in 1856, and he told the reader what every part of it meant. The goat head carries a torch between the horns for intelligence. One arm points up and one down, lettered with the words for dissolving and coagulating, to say that what is above and what is below answer each other. There are wings, a caduceus at the belly, a woman's breasts and a man's posture. Levi presented it as a figure of equilibrium between opposed forces, and specifically not as a devil.
From there it moved. Aleister Crowley took Baphomet as his own name within the Ordo Templi Orientis and read the figure through his own system. In 1966 the Church of Satan adopted as its sigil a goat's head fitted inside an inverted pentagram, a figure derived from a diagram published by the French occultist Stanislas de Guaita in the 1890s, and that sigil is now the most widely reproduced version of the idea.
Most recently The Satanic Temple commissioned a bronze statue of the seated figure with two children beside it, and offered it for installation at state capitols in the United States wherever monuments of the Ten Commandments stood on public ground, as a test of whether such displays could be restricted to one religion. The statue has been used in exactly that way in more than one state.
A name with no demonstrated medieval object behind it has therefore become a real presence in modern religion and in constitutional argument, entirely on the strength of a nineteenth century drawing.