The Jewish Year - a Calendar based upon both Solar and Lunar cycles
The Jewish, or Hebrew, Calendar Year
The Jewish calendar is based upon both the solar and lunar relationships to the earth. The Gregorian calendar (or the "secular" calendar as we know it today) is based on the yearly relationship of the earth and sun: 365 and 1/4 solar days make one year. The months are then divided more or less evenly between 30 and 31 days, except for February which is 28 or 29 days.
The Islamic calendar is based on the relationship of the earth and moon. You may have noticed that the Islamic holidsy of Ramadan appears at different times of year -- the solar year that is. That is because there is no compensation for the solar relationship to the earth.
The Jewish calendar is based upon both lunar and solar cycles with the earth. A "month" in the Jewish calendar is a lunar month. That is, the moon returns to its same relationship with the earth after 29 to 30 days. Jewish months are thus 29 or 30 days. However, you may note that 12 months of even a full 30 days does not fill a solar year. Due to Scriptural obligation that the Passover holiday always fall in the spring, a leap year is occasionally added. A Jewish leap year adds a whole extra month.
Results of lunar and solar relationship
Currently the Jewish calendar is a fixed calendar set by calculations. The month in which Passover falls (Nissan) always falls within a certain range of the solar year considered "spring". On the other hand, the beginning of a Jewish month is always just after the New Moon, when the first sliver of moon appears in the sky. If you've ever had trouble with your garden crops, check out an old Farmer's Almanac, and you'll see that it is the lunar cycle which is considered more important for crop success.
The "disadvantage" in today's work world, is that the Hebrew date slips around a bit every year with respect to the Gregorian date. So one year Rosh Hashana may be September 15th, and another year it may be October 3rd.
The Solution to the Jewish Calendar vs. Gregorian Calendar Dates
Many Calendars are available which show both the Hebrew date and the Gregorian date of each day. These are what are popularly called, "Jewish Calendars". In the Western countries, the months are usually split up according to the Gregorian months (January, February, etc.), with the Hebrew / Jewish month shown secondarily.
JewishByte.com publishes calendars through Cafepress.com, which only has regular Gregorian dated calendars. Jewishbyte has put original high resolution photos of Northern Israel on regular calendars. While this will not give you the Jewish dates and holidays on a calendar, the photos are beautiful and will bring you a touch of rural Israel.
There are a number of sources online carrying Jewish Calendars. This new year starts the year 5773. The new calendars out for 2013 will have the 5773 dates. Below is a small selection from Calendars.com:
For more information on the Jewish calendar and how it fits with the Gregorian calendar, check out these books:
The Structure And Mathematics of the Principal Calendars of the Western World: Muslim, Gregorian, Jewish, And Other Systems, and The Standard Guide to the Jewish and Civil Calendars: A Parallel Jewish and Civil Calendar from 1899 to 2050 With Parashiyyot and Haftarot and Candlelighting