This list is in no particular order. Feel free to comment!
The Ancient Greeks had sway via colonies and states from the Pillars of Heracles to India. The heart of Greece, though, provided the start of Greek greatness. Philosophers and their adoption of the scientific mindset allowed for rational thought and debate (though killing the philosopher was an all too common solution to science versus angry mob debates). Greek ideas of governance, culture, and religion were adopted and sometimes modified by the Romans who spread the culture throughout Europe, southwest Asia, and northern Africa.
Greek rationality was adopted by early Christians via Saint Paul and spread throughout the Roman Empire in a peaceful revolution based in part by Greco-Roman ideals. Greco-Roman culture was also adopted by barbarians like the Visgoths who continued Roman culture well after the Western Roman Empire's fall (a combination of good governance, religion, and the ability to make hot baths attracted many to Greco-Roman culture). The Eastern Roman Empire and its culture managed to keep its hold in eastern Europe until the fifteenth-century (if not longer because of the its partial adoption by the Russians). Even the conquering Ottoman Empire kept parts of the old culture including the title of Caesar.
Uniqueness: The Greeks gave the world philosophy, the modern understanding of science, and the ideal of democracy. The Romans gave the world the basis of civil law, which is the most popular form of law in the world.
The cultural hybrid of Norman French and Anglo-Saxon first took southeast Great Britain then the rest of the British isles. Being geographically isolated from continental Europe, a cultural theory of English uniqueness grew in many English minds. English looked at Troy for inspiration, civilized like the rest of Europe but separate and bordering on the wilderness.
The English and their subjects demographically colonize and dominate what would become the United States, Canada, Australia, and New Zealand. Not only this but other English colonizes adopted the democratic system (some more than others) and to this day follow many tenants of English cultures including using English as the national unity language and common law.
Unique Trait: The idea a person has rights that the state cannot control. The Magna Carta forced the state to recognize its limits. English later on expressed more rights. The start of the American Revolution, and many other revolutions thereafter, was based on the people being denied their God given rights by the state.
"There is no god but God and Muhammad is his Messenger" has been used throughout history as a cry for both war and peace. It has also been used as the cry to spread Arabic culture beyond the Arabian Peninsula. Along with Islam the Muslim advocates bring with them calls for Arab ideals and cultural norms. As Jewish culture goes with Judaism so does Arab culture goes with Islam.
Even the ideal of being an "Arab" has spread beyond true ethnic Arabs. There is a wide genetic difference between dark skinned Iranian Gulf Arabs, white Lebanese Arabs, brown Moroccan Arabs, and black Arabs of Saharan Africa yet all consider themselves to be Arab. Yet even those who do not consider themselves Arab are still deeply impacted by Arab culture through Islam, the Arabic language, or languages partially made by Arabs like Swahili.
Uniqueness: Besides the obvious of Islam, Arabic culture has been the only culture to replace either Roman or Roman-based cultures. Every other place once ruled by Rome still has a culture heavily impacted by Roman ways.
Coming Thursday: The last two cultures and special mentions