In 1869, the Russian chemist Dmitri Mendeleev arranged the 63 known elements by atomic weight in a table — and noticed that chemical properties recurred periodically as you moved across the rows. He left gaps in the table where the pattern predicted elements should exist but had not yet been discovered, and predicted their properties from their position. Gallium was discovered in 1875, scandium in 1879, germanium in 1886, all matching Mendeleev's predictions with eerie precision. The periodic table was vindicated as a deep regularity of nature, and chemistry had its first organizing principle.
The reason the table works was not understood until quantum mechanics in the 1920s. Elements have shells of electrons; the chemistry is dominated by the outermost (valence) electrons; new shells start at predictable atomic numbers, producing the periodic structure. The columns of the table reflect similar valence-electron configurations: the alkali metals (column 1) all have one loose s-electron, the noble gases all have a closed shell. The Pauli exclusion principle, the aufbau principle, and the quantum numbers governing electron orbitals all conspire to give the table its specific shape — eight columns in the main groups, ten in the d-block transition metals, fourteen in the f-block lanthanides and actinides. The modern table is organized by atomic number (proton count), not atomic weight, after Henry Moseley's 1913 X-ray work; this resolved the ordering anomalies in Mendeleev's original arrangement. Element synthesis extended the table from the 92 naturally occurring elements (uranium being the heaviest) into the transuranic actinides (neptunium, plutonium, americium, etc.) and now into superheavy elements with atomic numbers above 100 — most of which exist only for milliseconds before decaying. The current table extends to element 118 (oganesson, named for Yuri Oganessian, who is still alive and is one of the few people to have an element named after him in his lifetime).
Materials science — designing alloys, semiconductors, batteries, catalysts — is applied periodic-law chemistry. The current contests over lithium, cobalt, rare-earth elements are political competitions for materials whose properties are dictated by their position on the table. Computational chemistry, using density functional theory and increasingly machine learning, lets researchers predict properties of compounds without synthesizing them — a programme that has accelerated drug discovery and battery research substantially in the past decade.