In the nineteenth century, many chemists where trying to arrange the known elements into some sort of order based on the physical and chemical patterns that existed. What was mainly used in ordering the elements was their, at the time unreliable to determine, Atomic Mass.
Döbereiner came up with the idea of 'element traids' - groups of three elements with similar properties, which could also be linked to Atomic Mass.
Newlands proposed the idea of 'The Law of the Octaves'. This said that if the elements were grouped in eights and arranged in a table, similar patterns began to emerge.
Meyer further expanded the idea of 'Periodicity', publishing graphs of chemical properties against Atomic Mass.
The Modern Periodic Table
The chemist who created the table on which our modern version is based was Mendeleev. He built on the work of other chemists and created a table composed of Groups and Periods.
The order in which he positioned the elements was based partly on their Atomic Masses, although he switched some around such that they fitted batter with the observed patterns.
He left gaps in his table for elements he thought were yet to be discovered. He then predicted the properties of these elements, and when they were in fact discovered, and their properties matched his predictions, this aided in the widespread adoption of his table.
The credibility of his table was also helped by the fact that he did not rely entirely on the the then disputed Atomic Masses.