Astronomy is the basis for calendars, and the cycles of the sun and moon are the most important to the creation and understanding of most calendars.
For example, in our modern calendar, a tropical year (a tropical year is the time from one fixed point, such as the solstice and the equinox, to the next) is viewed as one revolution around the sun, which is equivalent to roughly 365.242190 days (this figure varies over time, as the year was 365.242196 days in 1900, an the year will be 365.242184 days in 2100).
A month is equivalent to a revolution to the moon (although our modern calendar rarely follows this pattern, but a month following the moon is called a synodic month). The rotation of the moon is 29.5305889 days (like the sun, this figure varies over time, as the synodic month will be 29.5305886 in 1900, and 29.5305891 days in 2100).
In reality, 29.5305886 is not the length of time for the moon to make one revolution around the earth, it is the length of time between two full moons. The diagram below should display why:
As you can tell, the distance the moon has to travel to be aligned up in the same position relative to the earth (to create a full moon in the example of the diagram), the moon has to travel a greater distance to take the extra angle into account. The actual time it takes for one rotation of the moon is 27.32 days. The equation below explains why:
Let the orbital period of the earth be Te = 365.25 days.
The time between full moons is Tf = 29.53 days
The angle, A, (in degrees) that the earth has moved in time Tf is 360Tf/Te.
The angle, B, that the moon has moved around the earth in Tf is 360 + A (see diagram, parallel lines subtend equal angles).
The orbital period of the moon is Tm, which we want to find. It took the moon Tf to move through an angle B, and it will take the moon Tm to move through a full rotation (360). The rate of angular rotation of the moon is constant, so these ratios are equal:
B/Tf = 360/Tm
So (360+A)/Tf = 360/Tm,
(360 + 360Tf/Te)/Tf = 360/Tm,
(1+Tf/Te)/Tf = 1/Tm
1/Tf+1/Te = 1/Tm
Tm = 1/( 1/Tf + 1/Te)
Tm = 1/(1/29.53+1/365.25) = 27.32 days
However, early calendrologist based the orbits of the moon on the time it took between two full moons / new moons / quarter moons / etc. As a result, the synodic month is based on this length of time, roughly 29.53 days.
Because the length of the tropical year isn't a multiple of the synodic month, the relationship between the sun and the moon can't be maintained. However, nineteen tropical years is equal to 234.997 synodic months, which is almost an integer. Because of leap years, the phases of the moon fall on the same date every nineteen years. This nineteen year cycle is referred to as the Metonic cycles, after the astronomer Meton of 5th century BC Athens.