In 1869, Russia's Dmitri Mendeleev and Germany's Lothar Meyer published nearly identical element classificaiton schemes, recognizing similar chemical and physical properties recur periodically when elements are arranged in order of increasing atomic weight.
Although their observations were nearly identical, Mendeleev is given more credit because his first published table predicted the existance of undiscovered elements, and left spaces in his table for them - even predicting their properties. Meyer left blanks in his second published table in 1870, not his first. Because of this, most of the world (except Germany) considers Mendeleev the Father of the Periodic Table.
In 1913, Henry Moseley developed the concept of atomic numbers - correctly stating that the atomic number is equal to the number of protons in the nucleus of an atom. The atomic number is also equal to the number of electrons surrounding the nucleus.
Arranging atoms by their increasing atomic number brought the elements in line with todays recognized periodic trends.
In 1945, Glenn Seaborg grouped the transuranium elements into the Lanthanide and Actinide series and proposed pulling them out of the main body of elements on the table.
Seaborg's arrangement does a better job grouping atoms on the Periodic Table according to quantum numbers.
The Periodic Table is based on the atomic theory. As the theory changes, so does the Table! The history of the Periodic Table traces our understanding of the atom.