It is no accident that several of these correspond directly to chemical series: the periodic table was originally created to organize the known chemical series into a single coherent scheme.
The modern explanation of the pattern of the periodic table is that the elements in a group have similar configurations of the outermost electron shells of their atoms: as most chemical properties are dominated by outer electron interactions, this tends to give elements in the same group similar physical and chemical properties.
The Roman numeral names are the original traditional names of the groups; the Arabic numeral names are those recommended by International Union of Pure and Applied Chemistry (IUPAC) to replace the old names.
There is considerable confusion surrounding the two old systems in use (old IUPAC and CAS) that combined the use of Roman numerals with letters. In the old IUPAC system the letters A and B were designated to the left (A) and right (b) part of the table, while in the CAS system the letters A and B were designated to main group elements (A) and transition elements (B). The former system was frequently used in Europe while the latter was most common in America. The new IUPAC scheme was developed to replace both systems as they confusingly used the same names to mean different things.
The periodic table groups are as follows (in the brackets are shown the old systems: European and American):