These ancient people believed the sun and the moon had powers of the gods, since both of these celestial objects were associated with happenings on earth (the sun, harvest, the moon, tides). The planets, which did not keep their positions realative to the other stars were also believed to have a direct effect on life on earth. Priests in Babylonia came up with the seven day week to honor the seven planet gods. The planet gods were believed to affect the rise and fall of kings.
Myths about the stars were generally stronger in the Americas. The people of Mesoamerica, like the Mayas and Aztecs, believed the stars affected everyday life and needed human help to continue. Some gods, such as Quetzalcoatl (the influence of the sun and Venus) had worship that included human sacrifice. In the cities, temples appeared to be aligned with objects of astronomical importance. In these cultures the study of astronomy affected their lives many ways. Transfers of royal power were timed with the summer solistice, monuments were aligned with astronomical objects, and the worship of astronomical gods. The people of the Maya civilization believed the gods needed human help and preformed many rituals they believed would guarentee the survival of the universe.
By having exact knowledege of the stars, rulers could keep absolute power by appearing "magical". King Menes of Egypt is an example since he could announce when the Nile would flood.
The Maya and Aztec people also studied the stars for calanders. The calanders they came up with were quite accurate, but extremely complex. The calender was based on a year of 365 days, with an approximate year of 360 days and a sacred almanac of 260 days. The sacred almanac was the most important. It was formed by all possible combinations of twenty names with thirteen numbers and each name and number was represented by a god. The astronomers were also priests, and they followed the motions of the sun and planets in "codices".
In Egypt, after watching the star Sirius, the Egyptians found the lenght of the years was 365 1/4 days. Thoth made a calander of twelve 30 day months with five extra days. The months were divided up into ten day units, and the day divided using decimal units. This type of calander was revived by France in 1795, but abandoned in 1806 because of the confusion it caused.
The Chinese kept records of the stars to predict solistices and eclipses. In ancient China they believed that an elipse was a dragon trying to eat the sun. They believed they had to make a lot of noise and beat gongs to frighten it away before it ate the sun. As a result, predicting eclipses was important to them. They used "gnomons" (pillars) to measure the shadow of the sun. They also observed the 12 year cycle of Jupiter.
The Greeks studied astronomy in depth, but they believed that the universe circled around the earth. Ptolemy developed a complicated model of epicycles. This was used to explain retrograde motion of the planets. Aristarchus of Sames countered the sun was much larger than the earth, so the earth must revovle around it. The Ptolemaic theorey was more widely accepted, however. Lucian, a Greek satirist, came up with an early idea of space travel- a fictional trip to the moon. When the Roman empire fell much of the Greek theory was lost to the Western world.
There are many possible markers of astronomical events. The pyramid of Khufu is aligned perfectly north-south. In the Americas, the Sun Dagger of the Anasazi Indians seems to have astronomical significance (at the summer solistice a ray of sun penetrates the center of a spiral), along with many Aztec ruins.