Here is a comparative table of some of the features of the Jewish and Chinese lunar calendars and holidays. The black and white circles represent the new moons and full moons throughout the year. The Jewish and Chinese new years are in bold. This is not a complete list of all holidays, but just enough to demonstrate how the calendars work. (For more detailed explanations, see notes below).
The Jewish and Chinese calendars use the cycles of the moon to determine the months. The Jewish New Year (Rosh ha-Shanah - literally, "the head of the year"), begins with the first new moon after the autumn equinox (September 23rd). The Chinese New Year (chun-jie - the "spring festival"), begins with the second new moon after the Winter Solstice (December 22nd). So the Jewish New Year (and the first day of the first month of the Jewish year) will always be sometime within the thirty days between September 23rd and October 23rd (as the first new moon could fall on any one of those days). Similarly, the Chinese New year (and the beginning of the first month) will always be sometime within the thirty days between January 20th and February 19th. In this way, the twelve lunar months are kept lined up with the seasons (determined by the sun in its course through the equinoxes and solstices). So, for example, the Jewish Passover will always fall in the spring, and the Chinese Lunar Festival will always fall in the autumn. Some Jewish and Chinese holidays will regularly overlap, such as Purim and the Lantern Festival (the fifteenth day after the Chinese New Year), both of which fall on full moons in the early spring, and the first day of Sukot and the Lunar Festival, both of which fall on full moons in the autumn. Periodically (in a leap year), an additional month is added to the calendars, to keep the twelve lunar months in line with the four solar seasons. (Notes on continuous yearly calendars below).
Notes on how the current Jewish and Chinese years are calculated:
The way the Jewish year is calculated today, the year 2010 is the Jewish year 5771 (which began in September of 2010). The way the Chinese year is calculated, the year 2010 is the Chinese year 4707 (beginning in February of 2010). The concept of keeping track of consecutive years from a single starting point was not originally used in the Jewish and Chinese calendars. The system of counting the Jewish year from the year of the "creation of the world" (in the fall of 3761 BCE, calculated by counting the generations listed in the Tanach back to Adam) did not become used until around the 3rd century CE. While the legendary Chinese Emperor Huang-di was credited with creating the Chinese calendar, starting from the beginning of his reign in 2697 BCE, this system did not come into common use until the last few hundred years. Prior to using these consecutive calendars, the Chinese would specify dates according to the various eras of Chinese history. Additionally, 60 year cycles were used. The year 2010 is within the 79th 60 year cycle since Huang-di. The Jews also had other systems prior to the consecutive-year Jewish calendar, such as specifying years according to the reign of a certain kingdom or empire. Additionally, there were 7-year cycles, which continued until a Jubilee year (Yovel in Hebrew) of 50 years (after the 7th 7-year cycle, or 49 years). The 50-year cycle would then start over again..
As a reference point, the year 1 CE would be the start of the Chinese year 2698, and of the Jewish year 3762. The Chinese year 1 would be the Jewish year 1065 (in other words, a Jewish year is 1064 years more than a Chinese year).