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).
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.
A few years ago, I was in a synagogue, and I overheard one man ask another, "When is Chanukkah this year?
" The other man smiled slyly and replied, "Same as always: the 25th of Kislev." This humorous comment makes an important point: the date of Jewish holidays does not change from year to year.
Years are either 12 or 13 months, corresponding to the 12.4 month solar cycle.
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.
Holidays are celebrated on the same day of the Jewish calendar every year, but the Jewish year is not the same length as a solar year on the civil calendar used by most of the western world, so the date shifts on the civil calendar.
The Jewish calendar is based on three astronomical phenomena: the rotation of the Earth about its axis (a day); the revolution of the moon about the Earth (a month); and the revolution of the Earth about the sun (a year).
The months drift around the seasons on such a calendar: on a 12-month lunar calendar, the month of Nissan, which is supposed to occur in the Spring, would occur 11 days earlier in the season each year, eventually occurring in the Winter, the Fall, the Summer, and then the Spring again.
On a 13-month lunar calendar, the same thing would happen in the other direction, and faster.
Note that most of the Bible refers to months by number, not by name.