/u/billbillbilly nails it pretty well.

The modern atomic theory didn't exist until the early 19th century, and things like conservation of mass, etc until the 18th century.

So forget the periodic table. "elements" were the four elements that all matter was made of: earth, air, fire, and water, going back to aristotle, except add a millenia of arabic commentators adding to the theory.

Like billbillbilly also mentioned, metal in 1500 was thought to be made up of mercury and sulphur. Salt minerals being needed to change the ratio. By changing the ration you change the metal.

Metals were thought to "ripen" in nature. This is based off of empirical evidence: there are traces of silver in copper ore, and traces of gold in silver ore, etc.

And so alchemists thought this process from lesser metals to precious metals can be done artificially. That's alchemy.

Basically you can break things down to their "elements" (base earth, air, fire, water) and then build them back up carefully into what you want (gold).

Alchemical recipes also have aspects of cabbalah, hermeticism, sympathetic magic, astrology, and all other sorts of mystyical aspects. See the History of Alchemy Podcast for a lot more.

Feel free to ask follow up questions if you want more details. You might also be interested in some of the other alchemy questions I've answered

The periodic table itself is also interesting.. but not direct to your question. In the 18th century chymists started noticing that some matter really can't be brocken down, not matter what. The new "atom" (the old "atom" was from Democritus and was really just theoretical)... at some point they noticed that these new elements had characteristics and things in common with each other. In fact three elements often had things in common with each other. These trifecta were some of the first patterns noticed that lead to the periodic table.