There are two very different meanings for a golden number, depending on what context the phrase is used in. A golden number in mathematics is usually used to refer to the golden ratio, a way in which two quantities interact. A golden number may also refer to a number given to a year to indicate where it falls in an astrological sequence, the Metonic cycle.
The Metonic cycle dates back to antiquity, when Meton of Athens, a famous astronomer, discovered that 19 years corresponded almost exactly to 235 lunar months. The Metonic cycle served as the basis for the Western calendar systems until the advent of the Julian calendar in the 1st century, and continues to be used in many different situations. A number of surviving calendar systems, most notably the Hebrew calendar, continue to be based on a 19-year cycle. Because twelve lunar months is substantially less than a true tropical year, the Metonic cycle included additional months in the 3rd year, 5th year, 8th year, 11th year, 13th year, 16th year, and 19th year of the cycle.
In modern times, the Metonic cycle is used infrequently, but does still get some use. It was, for example, used to calculate launch windows for the lunar missions. It is also most notably used to calculate the date of Easter for the Christian calendar, both in the Eastern Orthodox faith and in the Roman Catholic faith. This calculation, using the golden number and relying on the Metonic cycle, is known officially as the computus.
The golden number is a key part in determining the date of Easter, as it determines which year in the Metonic cycle we are currently in. It can be determined by the formula: Golden Number = X mod 19 + 1. To simplify, that means we take the current year in the Christian era, such as 2009, and divide it by nineteen, and then add one day. This extra day, known as the saltus lunae, helps correct the discrepancy in the calendar caused by a slight difference between the lunar months and the solar year.
The golden number is important in this calculation because Easter must fall on a Sunday within the third week of a lunar month that falls after the Spring equinox. This is a complex concept, and can be difficult to wrap your head around, so the golden number helps by simplifying the calculation of the month. In essence, the Spring equinox is determined by the solar calendar, and so falls on March 21st. At the same time, the traditional month system was based on lunar months, and Easter is determined based on this lunar system.
Finding the golden number lets you figure out where exactly the lunar month is in relation to March 21st. Once you know which lunar month has its mid-point, or formal full moon, after March 21st, you can then take that Sunday and determine it is Easter. To put it another way: within the solar calendar, which most of us use on a daily basis, Easter appears to drift around the calendar between March 22nd and April 25th. But in the lunar system, which the golden number represents, Easter always falls on the 3rd Sunday of the Paschal month, each and every year.