Please note, Henry has got the broad concepts sort of right, but the details WRONG. It is not the case that a leap year occurs every four years in the Jewish calendar. It is also not the case that after every leap year, the two calendars (Jewish and common) coincide.
The Jewish calendar is based on a lunar cycle (that is, each month is based on the interval, about 29 or 30 days, from new moon to new moon). The month begins at the new moon.
However, the lunar cycle of twelve months is therefore short by about 11 days per solar year. Since the Jewish holidays are supposed to be season-related (they were originally harvest festivals), the calendar adjusts by adding a whole month approximately every three years–actually, 7 times every 19 years, as Kyla quoted. The calculations are precise but VERY complicated.
A further complication is that the number of days in a month can change slightly (29 or 30)depending on exactly when the new moon occurs (like, if the new moon occurs shortly after sunset…)
There are other complications, so that (for example) Rosh Hashonah cannot fall on certain days of the week (like, Friday or Sunday) because that would impose severe difficulty on account of the interaction with Saturday (Sabbath).
In contrast, as Kyla notes, the Moslem calendar is also lunar, but does not adjust for the difference. The holidays are therefore not seasonal; while Ramadan is now in the winter (in the U.S.), it will shift slowly (by about 11 days each year) so that in thirty years or so, it will be in the summer (in the U.S.). Since Ramandan is a month of fasting during daylight hours, it is much easier now when the daylight hours are shorter than it will be in thirty years when daylight hours are longer.
I also should respond to Moe’s OP comment: << Why do the jewish holidays fall on different dates every year? >> …Well, that’s not so. The Jewish holidays always fall on the same day each year. Yom Kippur, for example, always falls on the 10th day of Tishri, and Passover always falls on the 14th day of Nissan.
BTW, in Israel, they tend to use both dates, Hebrew calendar and solar calendar (Gregorian).
Hope that helps.