Firstly we should understand that the months in Jewish law are determined according to the lunar cycle, as G-d said to Moshe when he showed him the new moon: “This month is for you the first of the months”. However the years are determined according to the solar seasons, since G-d commanded Moshe regarding the festival of Pesach: “Guard the month of spring”, meaning that Pesach should always fall in spring and not in any other season.
Since the seasons of the year are determined by the solar solstices, it follows that our calendar must be synchronized with the months following the lunar calendar and the years – the solar equinoxes.
If we calculate the lunar year we will find that it is 354 days, whereas the solar year is 365.25 days. Thus there is a discrepancy of about 11 days between them. Once in three years this discrepancy comes to 33 days, so we add another month to that year in order to coordinate the lunar and solar years. (Sometimes an extra day is added in Marcheshvan and Kislev to ensure the precise coordination of the two calendars).
Up until 1600 years ago, the way in which the lunar months were established was based on visual sightings of the new moon and similarly the decision whether to make a leap year was decided by the sages who would sit and discuss the issues involved.
This was the reason that they chose to add the month of Adar and not any other month, since it is the last month before Nissan and they wanted to wait as long as possible to see whether it was imperative to make a leap year or not. (There were other factors in play besides the need to coordinate the calendars). At a later stage Hillel the Nasi, a descendant of Hillel the Elder, established a fixed calendar which we use until this day.