It is the belief in the existence of the philospher's stone, that everything is made up from the same source essence and therefore "gold" the purest of all elements, can be transmuted from "iron" the most unpure. This is the lifelong goal of the alchemist, to discover the philosopher's stone by being capable of tapping into the source, through a mixture of science and magick.
There is also a lot of belief tied into planets, elements, alchemical symbols, and there is a HUGE philosophy and belief behind the practice. It is far too extensive to write out, and would take a book. And there are plenty out there. Check out sacred-texts.com for some. Particularly read Paracelsus. He was a genius and his writings are fantastic.
By discovering the philosopher's stone (Iron into gold) alchemists would prove their belief to be true. Most alchemists were and still are scientists. There have been claims of those who have discovered it, but they disappeared quickly, unfortunately. The most famous and successful alchemists lived during very Christian times. We all know how vicious Christians were at destroying anything or anyone that conflicted with their religion.
I think that most pagans share a belief in a "source", and hundreds if not thousands of years later, science confirmed that everything is made up of atoms! This is the key to the energy that every single object and living thing shares. The baseline of matter. I could go even further into detail about how the alteration of protons, neutrons and electrons within the atom cause transmutation! But all of this was confirmed way after alchemy was formed. I think the alchemists are onto something. =)
I personally enjoy all philosophy and remain open minded to the possibilities. I don't think that "all" of any philosophy or faith is 100% correct, but there are "truths" in most if you know how to look at them from the right perspective.
As WhiteRav3n points out, one goal of alchemy has now been realized - it's possible to turn lead into gold. (Though she is incorrect in one particular - it was lead, not iron, that was considered the basest of metals.) All you need to do to turn one substance into another is add protons, and enough neutrons and electrons to stabilize it.
But that's not how alchemists tried to do it; they were on the wrong track, as it turned out. Alchemy is still a viable philosophical system, but modern nuclear chemistry is what has finally let us transmute elements. However, there are other goals to alchemy, like the purification of substances, which may be of interest to you; I would search online for more information.
Now, the other ultimate goal of alchemy - immortality - still eludes us... but there's no reason to believe it's impossible. We age and die for several reasons, like the elimination of telomeres in chromosomes; there are many scientific techniques in the works to stop this from happening. Perhaps one day in the future, the philosopher's stone will be created by science.