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. 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). These three phenomena are independent of each other, so there is no direct correlation between them. On average, the moon revolves around the Earth in about 29½ days. 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, 29, 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. Years are either 12 or 13 months, corresponding to the 12.4 month solar cycle.
The lunar month on the Jewish calendar begins when the first sliver of moon becomes visible after the dark of the moon. In ancient times, the new months used to be determined by observation. When people observed the new moon, they would notify the Sanhedrin. When the Sanhedrin heard testimony from two independent, reliable eyewitnesses that the new moon occurred on a certain date, they would declare the rosh chodesh (first of the month) and send out messengers to tell people when the month began.
The problem with strictly lunar calendars is that there are approximately 12.4 lunar months in every solar year, so a 12-month lunar calendar is about 11 days shorter than a solar year and a 13-month lunar is about 19 longer than a solar 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.
To compensate for this drift, the Jewish calendar uses a 12-month lunar calendar with an extra month occasionally added. The month of Nissan occurs 11 days earlier each year for two or three years, and then jumps forward 30 days, balancing out the drift. 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!). Note that the traditional Chinese calendar follows the same sort of procedure, occasionally adding months to keep in sync, which is why "Chinese New Year" is always around late January or early February and Yom Kippur usually coincides with a Chinese fall festival (I'm not sure which one, but Chinatown is always very busy for the holiday the night when I break my fast!). The Islamic Hijri calendar, on the other hand, is strictly a lunar, 12-month calendar so the fasting month of Ramadan could fall in short, cool days of January (as it will in the late 2020s) or in the long, hot days of August (as it did in the early 2010s). An Islamic friend of mine once told me she preferred when it landed in January!
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. In English, we commonly call it a leap 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). Note that Adar II is the "real" Adar, the one in which Purim is celebrated, the one in which yahrzeits for Adar are observed, the one in which a 13-year-old born in Adar becomes a Bar Mitzvah. Adar I is the "extra" Adar.
In the fourth century, Hillel II established a fixed calendar based on mathematical and astronomical calculations. This calendar, still in use, standardized the length of months and the addition of months over the course of a 19 year cycle, so that the lunar calendar realigns with the solar years. 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). If you are musically inclined, you may find it helpful to remember this pattern of leap years by reference to the major scale: for each whole step there are two regular years and a leap year; for each half-step there is one regular year and a leap year. This is easier to understand when you examine the keyboard illustration below and see how it relates to the leap years above.
In addition, Yom Kippur should not fall adjacent to Shabbat, because this would cause difficulties in coordinating the fast with Shabbat, and Hoshanah Rabbah should not fall on Saturday because it would interfere with the holiday's observances. A day is added to the month of Cheshvan or subtracted from the month of Kislev of the previous year to prevent these things from happening. This process is sometimes referred to as "fixing" Rosh Hashanah. If you are interested in the details of how these calculations are performed, see The Jewish Calendar: A Closer Look.
The year number on the Jewish calendar represents the number of years since creation, calculated by adding up the ages of people in the Bible back to the time of creation. However, this does not necessarily mean that the universe has existed for only 5700 years as we understand years. 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"). 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.
Jews do not generally use the words "A.D." and "B.C." to refer to the years on the civil calendar. "A.D." means "the year of our L-rd," and we do not believe Jesus is the L-rd. Instead, we use the abbreviations C.E. (Common or Christian Era) and B.C.E. (Before the Common Era), which are commonly used by scholars today.
The Jewish calendar has the following months:
The length of Cheshvan and Kislev are determined by complex calculations involving the time of day of the full moon of the following year's Tishri and the day of the week that Tishri would occur in the following year. After many years of blissful ignorance, I finally sat down and worked out the mathematics involved, and I have added a page on The Jewish Calendar: A Closer Look, which may be of interest to those who want a deeper understanding or who want to write a Jewish calendar computer program. For the rest of us, there are plenty of easily accessible computer programs that will calculate the Jewish calendar for more than a millennium to come. I have provided some links below.
Note that the number of days between Nissan and Tishri is always the same. Because of this, the time from the first major festival (Passover in Nissan) to the last major festival (Sukkot in Tishri) is always the same (177 days). Likewise the fall holidays always begin two days later in the week than Passover.
Other than Shabbat, the name of the seventh day of the week, the Jewish calendar doesn't have names for the days of the week. The days of the week are simply known as first day, second day, third day, etc. Sometimes they are referred to more fully as First Day of the Sabbath, etc. Below is a list for those who are interested.
I maintain a current Jewish calendar on this website. Unlike most Jewish calendars you will see, my calendar shows the Hebrew months with the corresponding civil dates.
Most printed Jewish calendars cover a 16-month period: from September of one year (to include Rosh Hashanah) to December of the following year. Be aware, however, that some show only the 12-month period from September to August, and some that claim to have the full 16-month period show only limited information about September to December of the latter year. They show the civil months with Jewish holidays, Torah readings, candle-lighting times and so forth. I am particularly partial to the London Jewish Museum calendar, which has illustrations of Jewish artwork from the middle ages to the 1800s, but there are many Jewish calendars available on Amazon.com. Lately, I have been getting good Jewish calendars from local funeral homes, which is very convenient because they have the Shabbat and holiday candlelighting times for my local area right there on the date.
If you would like to look up the date of a Jewish holiday, from secular year 1 to secular year 9999, try HebCal. Of course, it is not precise for very early dates because at that time months were determined by observation, and earlier secular dates are artificial because the Gregorian calendar that we now use did not exist until the 16th century and was not accepted in many parts of the world until much later (they used the less accurate Julian calendar). Still, it is a very impressive piece of work. They also have a very nice, quick and easy converter to and from Hebrew dates on Chabad's website.