The name is older than the theology attached to it. In classical Latin lucifer simply means light-bearing, and as a noun it is the morning star, the planet Venus seen before dawn, the Greek Phosphoros. Jerome used it in the Vulgate to render the Hebrew helel ben shahar at Isaiah 14:12, in a passage mocking a Babylonian king who had aimed at heaven and fallen. In its own context the verse concerns a human ruler, and modern translations generally give day star or morning star rather than a name.
The step from that to the devil was taken by Christian interpreters who read Isaiah 14 alongside Ezekiel 28 and the line in Luke about Satan falling like lightning, and concluded that the passages described one event behind the visible one. Origen and later writers made the identification standard, and by the Middle Ages Lucifer was the devil's name before the fall and Satan his name after it. Luciferians take that same textual history and reverse the valuation attached to it.
The reversal usually runs through Eden. On the gnostic reading, the god who forbids the tree of knowledge is not the highest god but a lesser and jealous maker, and the serpent who urges eating from it is doing humanity a service. Lucifer is then identified with the impulse toward knowledge itself, and the fall becomes the moment humanity acquired the capacity to know and to choose. This is not a reading the biblical text supports on its own terms, and Luciferian writers do not usually pretend otherwise; the inversion is adopted deliberately and defended as such.
Beyond that shared core there is little agreement. Some Luciferians are theistic, treating Lucifer as an existing spiritual being who can be approached in ritual. Others are explicitly atheistic and use the figure as a symbol of reason, ambition and self-determination, which is close to how modern Satanists use Satan. Others again are broadly Jungian, treating Lucifer as an archetype of the individuating self. The tradition has no scripture, no founder and no body with authority to say which of these is correct, and any description of Luciferian belief is therefore a description of a range.