The Hadean Age is the earliest geological age in the earth's development, lasting from the time when planetary accretion was (mostly) finished, up until the time that the crust had (mostly) formed, which took roughly 500 million years. As could be guessed from the name, the Hadean was a time when the earth resembled some sort of underworld, with a molten sea of lava for a surface, temperatures too hot for water to form, and a poisonous atmosphere full of lightning storms.
Back when I was younger, I was a great reader of books on dinosaurs, which would usually pad their child-appealing diapsids with sections on the early and later eras of life. Geologically and biologically speaking, dinosaurs are much closer to us than they are to the truly archaic forms of life, but these books would still include a page or two on the earlier areas, such as the piscine Devonian or the bizarre Cambrian. And before then, there would usually just be a single illustration at the beginning, using mostly black and red colors, to describe the most primeval moment. Such pictures seem to have some type of primeval fascination for children (and others), who seek natural history as a way to suss out metaphysical questions about order and chaos.
The idea of the Hadean era certainly seems to fit in with early religious or cosmological ideas about the world forming from chaos and darkness, and it is probably as much from this impulse as from a scientific viewpoint that the Hadean gains its image. It is indeed, incredible and awe-inspiring to think about a time lasting a half-billion years, which is equal to the time since the first true multicellular life emerged until now, in which the world was without form, and void, and full of oceans of lava and constant rains of acid.
While this is a truly great picture, much less is actually known about what the Hadean was actually like. By definition, the Hadean is a time before we have a geological record, but whether this is because the earth was still molten, or whether most of those rocks were just eaten up by plate tectonics in the following billions of years is something that is hard to know--- although I am sure that there are legions of geologists who could correct me on the matter. It is known that there would be no oxygen, since there was no life, at least no life capable of photosynthesis. There may have been temperatures, at some point, low enough for oceans to form. The composition of the atmosphere, other than it would have probably have included carbon dioxide, methane, water, and other small molecules, would have been hard to guess. Many of the questions about how quickly the earth would have cooled down involve knowing how long it took for the supernova remnants that formed our proto-solar system to coalesce into planets. I have heard estimates as short as several million years, which would have allowed the heat of several shorter lived isotopes (those with half-lives on the scale of a million years) to contribute to the earth's heat. If the earth was formed over about 50 million years, the heat would instead come mostly from five much longer lived isotopes. Thus, trying to guess at the geological and chemical composition of the early earth involves having to guess at the timing of astronomical events.
While scientific research will produce more hints, such as microscopic fragments of rock from the earth's first few millions of years, it is probable that there will never be a very complete picture of what the Hadean period actually looked like. In the absence of such information, it will probably continue to be portrayed with the dark romance of lightning flashes over spewing volcanoes.