The Ancient Greeks' contributions to Western Civilization are so numerous that we can only really talk about a very few of them here.,
In mathematics, the Greeks contriubted to us the work of people like Eucild and Archimedes and Pythagoras in geometry and other forms of math. In the area of social sciences, Herodotus is known as the father of history as an academic discipline because he introduced the idea of a rational study of history (as opposed to a more mythological approach). Thucydides carried his work farther and is one of the most important historians ever. Ancient Athens, of course, is generally seen as the birthplace of democracy. We still look back to Plato and Aristotle as important thinkers on ethics and social issues.
All of these, and many more, are contributions to Western Civilization that have come from the Ancient Greeks.
Everything of which Western civilisation is composed today was based/developed in Ancient Greece. You name it... Chemistry, physics, astronomy, medicine, mathematics, language, literature, theater, music... The Greeks were the main contributors in the formation of the scientific world in which we live. Most names given to sciences and arts "physics, chemistry, biology, astronomy, theatre, etc" are of Greek origin (in fact they are anglicised versions of Greek words).
A few basic examples of the importance of the Greeks
1. Democritus came up with the idea of the atom (as you know this is the basis of physics and chemistry). Atom means "cannot be divided" and I think that speaks for itself.
2. Adaption of the Greek alphabet (αλφάβητος) by the Romans to form their own alphabet which as you know is today's "a,b,c,d..." letter used in most languages worldwide.
3. The basis of computing (Yes you read correctly) originates from ancient Greece with the invention of the Analog computer
4. Architecture evolved in Ancient Greece (Parthenon). Discovery of the Golden ratio
6. Because this can honestly go on forever; About (this is an estimation I read in a scientific magazine) 70% of scientific terminology originates from Greek, while most of the other terminology originates from Latin ( a language based on Greek, and formed by a people taught be Greek philosophers).
As you can see the Greeks contributed heavily to the world as we know it. In my opinion Ancient Greece is very rightfully labelled "The cradle of Western Civilisation" and I think that the Ancient Greece was the most important civilisation to ever walk on this planet....