The Greeks provided the general foundation for observational astronomy in western cultures. The western hemisphere uses their names and mythology for constellations (for the most part).
The Greeks also observed planetary motion, naming them after the gods.
Aristotle, a greek philosopher, laid down various tenets about the natural world that were studied and upheld for nearly two thousand years before the rise of modern science demonstrated them as inadequate and inaccurate.
Most cultures have their own constellations, observations and mathematics that overlap the Greeks. However the Greek ties with Rome, and the rise of the Roman Empire made their contributions the most prominent in the western hemisphere.
The Greeks, by observing the earths shadow on the moon, and way ships disappear over the horizon of the ocean, determined that the earth was round. One significant contribution that Greeks provided was observations by Aristarchus (~300 BC) estimated (fairly accurately) the diameter of the earth by using shadows cast down a well in two different cities.