Before the periodic table, there were a bunch of symbols, number, letters etc (In all kinds of languages) that represented the elements. Scientists around the world saw that a chart of the elements needed to be universally accepted and finalized. A guy named Mendeleev presented this idea to the scientific community. Mendeleev was also the first to order elements according to atomic number rather than atomic weight. The modern day periodic table was not published by him, it was developed with the help of the entire scientific community. Honestly, there isn't a specific way to tell you how the periodic table was constructed, scientists developed thousands of tables that represented the elements. And just to let you know, the modern day periodic table is constantly going through changes as we discover more and more about elements, atoms, molecules etc. so in the near future it wouldn't be surprising if we saw something completely different than what we see today.
