There are 12 months on the Jewish calendar. A month is the period of time between two renewals of the moon. Since a complete cycle of the moon is about 29 days, the entire year in the Jewish calendar consists of 354 days.
Dates are very important in Judaism. The holidays must be celebrated at a specific time and during a specific season. The seasons change based on the position of the sun throughout the year. A solar year - a full cycle of Earth around the sun - is 365 days.
There is an 11 day difference between the solar year and the lunar year. That difference may cause the seasons to shift a little ever year so they won't occur during the same months every year.
To avoid that, every third year is called a leap year. A leap year is a year of 13 months instead of 12. The added month compensates for the 11 day difference and helps keep the months aligned with their respective season.
The Torah does not call the months by name. The Torah relates to the months based on their position in the year - 1st, 2nd etc. Later on, the Jewish sages had given each month a name, names we use until today.
On the first month, Tishri, the Jewish people celebrate three holidays: Rosh HaShana, Yom Kippur, Sukkot. On the third month, Kislev we celebrate Hannukkah. On the fourth, Tevet, we mark the day in which the siege on Jerusalem had started - a siege that led to the destruction of the Temple.
On the fifth, Shvat, we celebrate a new season ('new year') for the plants.
On the 6th, Adar, we celebrate the holiday of Purim.
On the 7th, Nisan, we celebrate the holiday of Pesach (Passover).
On the 9th, Sivan, we celebrate the receipt of the Torah, by Moses and the entire Jewish nation at the mountain of Sinai.
On the 10th, Tamuz, we mark the fall of Jerusalem.
On the 11th, Av, we mourn the destruction of the two Temples - both were destroyed on the same day.
Every month, as the new moon appears, we celebrate the start of the new month.