Adar I is added in the 3rd, 6th, 8th, 11th, 14th, 17th and 19th years of the cycle.The current cycle began in Jewish year 5758 (the year that began October 2, 1997).
In ancient times, this month was added by observation: the Sanhedrin observed the conditions of the weather, the crops and the livestock, and if these were not sufficiently advanced to be considered "spring," then the Sanhedrin inserted an additional month into the calendar to make sure that Pesach (Passover) would occur in the spring (it is, after all, referred to in the Torah as Chag he-Aviv, the Festival of Spring! A year with 13 months is referred to in Hebrew as Shanah Me'uberet (pronounced shah-NAH meh-oo-BEH-reht), literally: a pregnant year. The additional month is known as Adar I, Adar Rishon (first Adar) or Adar Alef (the Hebrew letter Alef being the numeral "1" in Hebrew).
The extra month is inserted before the regular month of Adar (known in such years as Adar II, Adar Sheini or Adar Beit).
This process is sometimes referred to as "fixing" Rosh Hashanah. However, the Jewish New Year is in Tishri, the seventh month, and that is when the year number is increased.
If you are interested in the details of how these calculations are performed, see The Jewish Calendar: A Closer Look. This concept of different starting points for a year is not as strange as it might seem at first glance.
Many Orthodox Jews will readily acknowledge that the first six "days" of creation are not necessarily 24-hour days (indeed, a 24-hour day would be meaningless until the creation of the sun on the fourth "day"). The names of the months of the Jewish calendar were adopted during the time of Ezra, after the return from the Babylonian exile.
For a fascinating (albeit somewhat defensive) article by a nuclear physicist showing how Einstein's Theory of Relativity sheds light on the correspondence between the Torah's age of the universe and the age ascertained by science, see The Age of the Universe. The names are actually Babylonian month names, brought back to Israel by the returning exiles.
The Earth revolves around the sun in about 365¼ days, that is, about 12.4 lunar months.
The civil calendar used by most of the world has abandoned any correlation between the moon cycles and the month, arbitrarily setting the length of months to 28, 30 or 31 days.
The Jewish calendar, however, coordinates all three of these astronomical phenomena.
Months are either 29 or 30 days, corresponding to the 29½-day lunar cycle.