There are several aspects to the organization of the periodic table of the elements.
First, all the elements are ordered, going left to right, by atomic number – which is the number of protons in the nucleus of an atom of that element. So, the first is hydrogen, with atomic number 1, then next is Helium with atomic number 2, etc.
Next, there are rows which relate to the quantum mechanical theory of electron shells. A new row starts in the table when the previous shell has been filled.
The vertical organization is usually considered the most important, since the blocks formed by columns reveal many of the chemical properties of the elements in that group. Specifically, elements are grouped in s, g, f, p, and d blocks according to the subshell in which the ‘last’ electron resides.
The columns in the Periodic Table, describing elements with similar chemical properties, group elements that have the same electronic configuration in their outer shells into a family; the rows in the table, like an octave in a musical scale, determine a period or a set of elements, with each member representing one of those families. The rows are not categorized strictly on the basis of atomic weight, but on similar properties of elements within families. In general, the 1st ionization energy (1st ie), the amount of energy needed to remove an element's most energetic electron (usually the outermost electron, but not always) increases going bottom to top (up the family) and left to right (across the period).