The Jewish calendar is based on three astronomical phenomena: the rotation of the Earth about its axis (a day); the revolution of the moon about the Earth (a month); and the revolution of the Earth about the sun (a year). These three phenomena are independent of each other, so there is no direct correlation between them. On average, the moon revolves around the Earth in about 29½ days. The Earth revolves around the sun in about 365¼ days, that is, about 12 lunar months and 11 days.
To coordinate these three phenomena, and to accommodate certain ritual requirements, the Jewish calendar consists of 12 or 13 months of 29 or 30 days, and can be 353, 354, 355, 383, 384 or 385 days long. The keystone of the calendar is the new moon, referred to in Hebrew as the molad.
A new month on the Jewish calendar begins with the molad, (pronounced moh-LAHD). Molad is a Hebrew word meaning "birth," and refers to what we call the "new moon" in English. The molad for the month of Tishri (the month that starts with Rosh Hashanah) is the most important one for calendar calculations, and is referred to as Molad Tishri.
Note that the calculated molad does not necessarily correspond precisely to the astronomical new moon. The length of time from one astronomical new moon to the next varies somewhat because of the eccentric orbits of the Earth and Moon; however, the moladot of Rabbi Hillel's calendar are set using a fixed average length of time: 29 days, 12 hours, and 793 "parts" (or in Hebrew, chalakim). The amount of time is commonly written in an abbreviated form: 29d 12h 793p.
A "part" (or in Hebrew, cheilek) is a unit of time used in the Jewish calendar, equal to 3-1/3 seconds. There are 18 parts in a minute and 1,080 parts in an hour. Most sources express time from calendar calculations in days, hours and parts, although some sources break the parts down into minutes. For example, the period between moladot could be written as 29 days, 12 hours, 44 minutes and 1 part (29d 12h 44m 1p), because 793 parts is 44 minutes and 1 part (793 = 44 times 18 parts plus 1 part) . This makes the resulting times look somewhat more familiar, but it increases the number of calculations, so we will stick with days, hours and parts.
The same shorthand can be used to express the time when a molad occurs. The time is normally expressed as a day of the week, along with the hours and parts (or hours, minutes and parts). For example, the time of a molad might be expressed as 2d 12h 1005p (or 2d 12h 55m 15p), meaning that it occurs on Monday (the second day) at the 12th hour and 1005 parts.
The "hours" used to calculate the molad are standard 1/24 of a day hours. Note that this differs from the "hours" used for ritual scheduling, which are 1/12 of the time from sunrise to sunset. For example, at Pesach (Passover), we are required to stop eating chametz at the end of the "fourth hour "of the morning on Nissan 14, that is, at the end of 1/3 of the time between sunrise and sunset. These "seasonal hours" vary depending on the time of the year; molad hours are constant. The time for the molad is Jerusalem Solar Time, which is not necessarily the same as your local time. It is also not necessarily the same as the time on the clock, even in Jerusalem. This fact has no effect on your calculations, but is worth knowing.
The Jewish "day" starts at sunset, rather than at midnight. If you read the story of creation in Genesis Ch. 1, you will notice that it says, "And there was evening, and there was morning, one day." From this, we infer that a day begins with evening, that is, at sunset. Accordingly, most sources discussing the molad use 6PM of the preceding evening as the "zero hour." In our example, 2d 12h 1005p, the 12h means the 12th hour after 6PM, that is, 6AM. If a molad occurs at 2d 4h 0p, this means that it occurs at 10PM on Sunday night, because the second day (Monday) begins at 6PM of the preceding evening (Sunday). Some sources, however, use the more familiar Western conventions and use midnight as the zero hour. Be very careful to check which system is being used when you rely on times given by any source! If the time is referred to as "Rambam time" or something similar, then you know it uses 6PM as the zero hour. On this page, I am using Rambam time, but some well-respected Orthodox sources in America use midnight as their zero hour. As long as you are consistent, you will get the same result under either system.
The above gives you a general idea of the complications of Jewish calendar calculation, but will not enable you to calculate a specific date, or to express a specific western date as a Jewish date, or determine the specific date of Rosh Hashanah or Pesach in terms of the Western Calendar.
The detailed information on how the calculation is done can found on this page of Judaism 101 which I have used as an excellent source for the other calendar information on this and the previous page.
For detailed information on when the Feasts and Fasts occur in the Jewish Calendar Cycle, go to Calendar Cycle View also accessible directly from the left panel Main Menu.