Structure of the periodic table
Mendeleev was the first to realize that properties recur among the elements. To highlight this, he devised the early periodic table based on atomic masses. Moseley came along and stated that the periodic properties of elements was a function of their atomic numbers rather than their atomic masses. 



It was Niels Bohr who came up with the idea of shells and subshells. He developed his own take on the periodic table, which looked somewhat like this:



Finally, coming to your question, A lot of derivatives of the periodic table were made by various scientists during the 1920's - 1940's. The version we use today was made by Glenn T. Seaborg and his colleagues.


  Seaborg and collaborators had synthetically produced several new elements with atomic numbers beyond uranium, the last naturally occurring element in the table. Seaborg saw that these elements, the transuranics (plus the three elements preceding uranium) demanded a new row in the table, something Mendeleev had not foreseen. Seaborg’s table added the row for those elements beneath a similar row for the rare earth elements, whose proper place had never been quite clear, either.


Further reading:


How the periodic table went from a sketch to an enduring masterpiece
Further development of the Periodic Table - Wikipedia