What area did they live in? 
Ancient Greeks lived on the Balkan Peninsula (approximately at the territory of modern Greece) and on the western part of Turkey. They governed themselves from their cities, each of which had a separate government which governed over the countryside surrounding them. There were quite a few of these city states, but their two greatest cities were Athens, in the north, and Sparta in the south. The Greeks lived under their own rule until first being absorbed by the Macedonians, who lived to their north, and then finally the Roman Empire. Greece has many mountains and poor soil, which prevents many crops from being grown there. The country has a mild climate, which makes it convenient to farm goats and sheep. Also, Greece is a sea country: there are many beaches and islands, and no part of the country is far away from the sea. That made many of the Greeks sailors, and that meant that they sailed near and far, trading with different people, learning about other cultures, and bringing their own culture to faraway lands. These types of exchanges inspired development of Greek science and art.
What did their structures look like? 
Greek buildings were usually rectangular-shaped, surrounded by colonnades – rows of columns. Greece is rich in limestone, which was the most popular material of the time. Marble was more expensive, and was mostly used for statues and decoration.
Greek theaters looked very much like modern ones, except that most were open to the weather. There were semi-circled rows of seats, each row was higher than the row in front of it, so that people in the back could see better. Centered in front of the audience was a circular orchestra, where performances took place. A low building called a skene sat behind the orchestra. It was used to conceal actors, and gave room for costume changes, props and other "backstage" activity.
What did they wear? 
Greek men wore chitons – pieces of cloth pinned on one shoulder. Women wore peplos – long tube-shaped cloth, which was pinned on both shoulders and gathered around the waist. When it was cold, both men and women wore a cloak, called clamys. Greeks wore sandals, which looked a lot like the sandals that we wear today.
What did their writing look like? 
Greek writing looked a lot like Modern Greek text, written with capital letters. The Greek alphabet appeared in 8th century BC and was influenced by the Phoenician alphabet, the first alphabet that used signs for letters, not for words or syllables. Unlike the Phoenician alphabet, which did not have characters for vowels, the Greek alphabet had characters for all the sounds used in the language. In that sense it was the first modern alphabet, and the oldest to be used to this day. The Greek alphabet became a base for two other alphabets: Latin (that is used in English) and Cyrillic currently used in Slavic-speaking countries.
What did they believe? 
Greeks were polytheistic, which means that they believed in many gods and goddesses. Each god or goddess was “responsible” for some side of life or natural phenomenon.
Twelve of the most important gods resided on mount Olympus. Their leader was Zeus, the god of the sky. Other Olympic gods were: his wife Hera - the goddess of marriage and motherhood, Poseidon – god of the sea, Aphrodite - the goddess of love, Apollo – the god of art and leader of muses - and Athena, the goddess of wisdom and war, and the favorite daughter of Zeus.
Greeks believed in many mystical creatures other than gods: titans – children of Gaia, goddess of Earth, challenging Olympic gods; nymphs – nature spirits; muses – goddesses and patrons of art; centaurs – half humans, half horses; Cyclops – one eyed monsters; giants.
In Greek myths, the Gods often had affairs with real people, and that’s how heroes, like Hercules – man of superhuman strength, who committed twelve labours, and Achilles – the hero of Trojan war, were born.
Are some of them famous even today? 
We know many Greek politicians, scientists, and artists. Possibly the most known person of this culture is Homer, the legendary blind poet, who composed two masterpieces of Greek literature: the poems Iliad and Odyssey. Sophocles and Aristophanes are still popular playwrights and their plays are considered to be among the greatest works of world literature. Another famous Greek is a mathematician Pythagoras, mostly known for his famous theorem about relations of the sides of right triangles. Archimedes made many amazing inventions and discoveries. The legend says that he discovered Archimedes Principle while taking a bath, and got so excited, that he ran out naked into the street crying “Eureka!” – “I have found it!” The philosopher Socrates taught to question everything, separating beliefs from proven facts. He even questioned the existence of gods, which got him in trouble: he was accused of corrupting the youth and was sentenced to death by taking poison. Other great philosophers include Aristotle (the teacher of Alexander the Great) and Plato. Alexander the Great is known for his conquest of the Middle East and Central Asia.
What is left of them today? 
The whole modern country of Greece! They still speak the same language, though it has changed some, use the same alphabet and preserve their heritage.
But Greek heritage is not limited to Greece only. It influenced Western culture greatly. That's why we call it the Cradle of Western Civilization. We use many Greek words in our everyday life, like democracy, philosophy, photography. In fact, the word alphabet comes from alpha and beta, the first two letters of the Greek alphabet. In mathematics, one of the greatest discoveries they made was about right-angled triangles which is still important in map making. Greek philosophers strongly influenced ideas in the West for many years and their ideas are still important in politics, religion, government, discussion and argument, and in how people should behave toward one another and in society as a whole. Greek architecture and art strongly influenced Western architecture and art. (What does Lincoln Memorial remind you of?) The foundation of our science came from Ancient Greece. The Greek style of sculpture is still used all over the world today for statues of famous people. The Greeks also invented the theater and many of their plays are still acted all over the world today. Many of their stories and poems are also read widely today. The Greeks were the ones who started the Olympic Games, back in 776 BC, in honor of the god Zeus at the town of Olympia. Soon their fun side became more important and they were not just religious festivals anymore. Games were held every four years until AD 394 and were later restarted in 1896. The first of the new Olympics was held in Greece in honor of their original home and they have grown to become popular worldwide.