Alchemy is arguably the important forerunner of modern chemistry although many will be familiar with it for the rather less auspicious association of trying to turn base metals into gold. This now defunct interdiscipline was practiced throughout the Classical world and continued in Europe right up to the 19th Century until it was finally wholly supplanted by modern scientific theory and methodology. Alchemy combined a variety of different approaches to nature taken from the physical sciences, medicine, and semiotics, but also from non-scientific disciplines such as astrology, mysticism, and spiritualism.
When most people think of alchemy they think of its typical portrayal in popular culture of bearded old men working frantically over pots, pans, beakers, and cauldrons filled with all manner of ingredients, in the pursuit of a method of creating gold instead of having to dig it up. But this was just one of a variety of goals that the alchemists pursued. Another of these was the search for a panacea, a way of curing all illnesses and extending life indefinitely. A further aim was to find a universal solvent. These are, of course, all aims that have been pursued by modern scientists as well.
The difference between the alchemists and the later modern scientists (although some people, even famous figures like Isaac Newton, who had a foot in both camps) is in the methods used by the two groups. The alchemists, instead of using the scientific method, relied on a belief in mystical ideas that would allow them to harness some underlying spiritual force to achieve their ends. In particular they searched for what they called the Philosopher's Stone, a magical substance that would achieve the transmutation of metals or help create a panacea.
Whilst the search for the Philosopher's Stone and the alchemist's mystical worldview in general were to prove fruitless, some of the side effects of their activities proved useful to modern science. Alchemists provided contributions in various areas of the physical sciences, producing advances in gunpowder, paint, ink, and many other things.
Ironically, the atomic theory was later to show that the nuclear transmutation of elements into each other is indeed possible, meaning that if you really wanted to you could, in fact, turn a lump of lead into a lump of gold. This method is very different though from what the alchemists were thinking of and also doesn't satisfy their real motivation for doing it in that it can't be done in a way that would make you a profit.