It is based on the idea that the world is composed of the four elements fire, air, water and earth. These elements are considered to be contained in all things, the most common example being wood which "obviously" contained earth (which was why it was hard) and water (sap when squeezed out) and air (smoke when lit) and fire (you could light it). It was a nice little theory...
The basic idea of the atom as the "smallest unit of divisibility" was also formed during this time - though this truly bears little relation to the present-day concept of the atom (which is merely named after the Democritian concept). Atoms were considered to be tiny, perfect spheres and made up of one of the four above-mentioned elements... like I said, a nice little theory and a lot more advanced than the infinite-turtles style of science that was around previously.
One of the concepts was that, because lead and gold (the perfect metal) were from the earth element, earth could therefore be turned into lead and gold using an "appropriate catalyst" known as the Philosopher's Stone. This was a magical substance which could turn water into wine, heal the sick, and as mentioned, turn really boring lumps of metal into gold and silver. Most of it though was just processes like mixing copper with arsenic to give it a silver finish, or a philosophical trick by which the alchemist's apprentice, through years of hard work and training, was turned from lead (i.e. an ignorant klutz) into gold (i.e. an alchemically educated klutz). :)
The quest to turn lead into gold was also a spiritual one, as it represented the change of an impure substance into a pure substance - the equivalent of returning Man from his current state of sin to a state of Grace last known before the Fall.
Despite never discovering the Philosopher's Stone, alchemists did come up with a number of interesting reactions and the knowledge gathered eventually led to the foundations of modern chemistry.
Alchemy also gave rise to concept of the panacea, the "medicine" to cure all illnesses - also never found, but a search for it led to an interesting number of ways to poison oneself (mercury, silver and lead poisoning being numbered under the chief ways that alchemists died).