Adar II is the additional, thirteenth month inserted into the Hebrew year in a leap year, falling toward the end of winter and counting thirty days. The Hebrew calendar is lunisolar: twelve lunar months fall about eleven days short of the solar year, so in seven years out of every nineteen a whole extra month is added to keep the festivals from drifting through the seasons, and it is Adar that is doubled. In a leap year the first Adar becomes an ordinary month and Purim is kept in Adar II, on its fourteenth, so that the festival stays in the month adjoining Nisan and Passover. This intercalation is what holds Passover in the spring, as the Torah requires. These are the features of the Jewish calendar tradition, in which Adar II is the mechanism that reconciles the lunar months with the solar year.
Adar II exists only in leap years, seven times in every nineteen-year Metonic cycle used to fix the Hebrew calendar, and in those years it, not the preceding Adar I, carries Purim on the fourteenth and the fast of Esther on the thirteenth, along with all the other observances ordinarily associated with the single month of Adar in a regular year. The doubling of Adar rather than of some other month reflects a specific halakhic concern: inserting the extra month here keeps Nisan, and with it Passover, anchored to spring, since Passover's biblical date is explicitly tied to the barley harvest and must not drift into winter, the very problem intercalation exists to prevent across the whole lunisolar Hebrew calendar.