In the 1860s Mendeleev arranged the elements in order of atomic weight, which is now called relative atomic mass, and he put elements with similar properties in the same column. He left gaps in his table for elements that had not yet been discovered and predicted their properties. When the missing elements were found, his predictions turned out to be correct, and scientists accepted his table. Where atomic weight order put an element in a group that did not match its properties, he swapped the order of some elements, such as tellurium and iodine, so that iodine sat with the similar halogens.
Later, scientists discovered the proton and the nucleus. This led to the idea of the atomic number, which is the number of protons in the nucleus. The modern Periodic Table is arranged in order of increasing atomic number. This removes the need to swap elements. For example, argon has a higher relative atomic mass than potassium, but argon has atomic number 18 and potassium has atomic number 19. Ordering by atomic number puts argon before potassium, with the noble gases, where its properties fit.
Atomic number also explains the layout. In a neutral atom the atomic number is the number of electrons, so it sets the electron arrangement. Elements in the same group have the same number of outer shell electrons, which is why they have similar properties. The period number is the number of occupied shells. The noble gases were discovered after Mendeleev, and they were added to the table as a new group.