Dmitri Mendeleev and Julius Mayer published very similar tables independently of one another. Mendeleev is commonly credited with the development because his version was published in 1869, while Mayer's version was published in 1870.
Mendeleev's table of elements was much better than previous attempts for several reasons. First, his table left gaps for undiscovered elements, predicting the properties of the elements based on their location within the table. Second, Mendeleev chose not to sort the elements by atomic mass, (which was the common practice) instead choosing to base it solely on the properties of the elements. When atomic structure theory was developed, it was later revealed that Mendeleev had successfully sorted his table by atomic number (the quantity of protons present in a given element).