European shipyards in 1500 CE were used to manufacturing small, fast vessels capable of sailing into the wind. They did it all the time, and in fact such vessels had existed for quite a long time, going to and fro across the North Sea and Mediterranean. European ships were dwarfed by the Chinese trading/tribute vessels of the fifteenth century; while they were matched in seaworthiness, the Chinese ships could carry far more people and far larger and more impressive goods to trade. Cristoforo Colombo sailed to the Caribbean in what Chinese sailors would have seen as little more than a river ferry. But what is important is that he got there, not the size of his vessel.
Could Chinese sailors have reached the Americas? The lunatic pseudohistorian Gavin Menzies has his own answer to this question, claiming that they actually did reach the Americas, and Australia to boot, before Europeans got there. But this is a fraudulent claim. Chinese sailors certainly reached east Africa, India, and most of maritime southeast Asia in the fifteenth century. They even extracted tribute from Bengal and had Sri Lankan kings taken to Beijing as punishment for transgressions. What they did not do is reach America.
Well, why would they? The spice islands were to the south, not the east. Other treasures came from the west, across the Eurasian continent through the Yunnan-Myanmar trading route, the so-called 'Silk Road' through central Asia, and the 'maritime Silk Road' linking the Indian Ocean in a vast system of mercantile activity. Nothing came from the east; as far as the Chinese were aware, and the Japanese too, there was nothing to the east. There was no incentive to go east. And the vast, frightening, murderous 'Pacific' Ocean prohibited accidental contact between east Asia and the Americas. They had the technology to traverse the Pacific but no real reason to do so.
Europeans, however, were well aware of the spice trade and where it came from; it came from the east, far across the sea and with at least one, if not two, continents in between. They were aware of trade in the Indian Ocean - in fact, many Europeans powers, including Venice and Ragusa, were active participants (at the battle of Diu in 1509, the Portuguese fought a coalition of Venice, Ragusa, the Ottoman Empire, Kozhikode, Gujarat, and several other powers). They knew the earth was a sphere and believed that the Atlantic Ocean probably went all the way around to the spice islands. They had honed their ships' fighting capabilities in the harsh geopolitical realm of the Med. They had every reason to go south and west in search of spices, and thereby almost accidentally discovered the Americas (as well as a seaward route under Africa following the prevailing winds by shooting out west into the Atlantic, an idea first proposed by Bartolomeu Dias).
Europe's lack of political unity encouraged competition. This, combined with high population density and a high level of literacy in a common language, Latin, gave impetus to European technology. It is notable that the cannons fired at the siege of Constantinople in 1453 were manned by mercenary European crews. Lack of unity also gave individual kings and queens the ability to make decisions about their investments, and it meant that navigators could sell their services to the highest bidder. Fernao de Magelhaes transferred his services from Portugal to Spain when Spain paid him more. Colombo, a Genovese, went to several nations for investment, finding most support in Spain. This was not possible in China, where the 'treasure ships' were halted by imperial bull. The political divisions of Europe were enabled, but not totally determined, by the fragmented geography of the continent. China could be, and was, conquered early by local conquerors, reducing the internal competition of the area. A large part of China proper is plain. Europe is cut into quadrants by mountains and small seas.
Europe was also highly literate and had been for some time. Literacy was in many hands, not just the clergy or government, something again encouraged by political division. Rich merchants could learn to read and write in Latin or their vernacular languages - Marco Polo's account of his thirteenth-century travels in Asia was written in Old French - and books were relatively cheap as a result of the introduction of paper from China hundreds of years before and the (probably independent) invention of printing in Germany. This meant that innovations and ideas could be propagated across the continent with ease. There was no reliance on oral transmission and there were plenty of buyers out there for information about the Indian Ocean spice trade. Ancient texts, including the best of the Greco-Roman geographic tradition, were still around and were frequently copied. Ptolemy, Strabo, Eratosthenes, the Periplus Maris Erythaei, and any number of ancient maps - they were all well-known to European navigators and thinkers in the fifteenth-century CE.
Europe was also perfectly placed to 'benefit' from the diseases of Eurasia. In this sense it was no different from the rest of the continent, except that European motivation to sail west meant that Europeans were the first Eurasians to reach the Americas. This meant that they were the first to introduce Eurasian diseases to American populations - and this gave them the edge in extracting wealth from the continent. This went into fueling the European economy for hundreds of years, making Europe far richer than anywhere else in Eurasia. This meant that Europeans could fight unprofitable wars, because they had enough money to burn, and it meant that inter-state competition was even greater. This, in turn, led to technological advances, and thereby yet more conquest of far-off places, and thereby more money to burn, more competition, more technological advance, more conquest, and... tah-dah, the world is European.
The answer to the question of why Europe, rather than, say, China, conquered the world is reasonably clear, and it is due, at least in part, to geography. Geography supplied the means: the spread of agricultural technologies by the east-west orientation of Eurasia; the rise of inter-continental empires through the use of the horse, camel, and sailing vessel; the propagation of murderous diseases and sophisticated technologies, including paper and the compass, by both of these causes; and so on. The motivation was produced, indirectly, by the same geography; it wasn't easy for Europeans to get access to the spice trade, so they had to come up with a way to have at it. They came up with several ways. They then used the fruits of this - including their domination of the Americas - to improve their technologies and one-up everybody else.
There is a common misconception that in the year 800 CE, Europe was a backwater. It was not. There were two reasonably powerful empires in existence, the Carolingian and Byzantine. The population was large, especially when compared to, say, Africa in the same period. Literacy was high - even if Charlemagne couldn't read, many of his subjects could, and he encouraged the rise of literacy across his domain. In Anglo-Saxon England, there was an enormous amount of money and there, again, literacy was high. There were lots of population movements going on, and the end of a number of kingdoms, including Visigothic Spain, but the idea that this was a 'dark age', as it is known in popular historiography, is simply wrong.
Latin never died in Europe, nor did vernacular literacy. Urbanisation didn't disappear, either. If Europe was in a 'dark age', or was simply a Eurasian backwater, in 800 CE, then tropical Africa has never seen a 'light age'*, and southeast Asia has always been in a 'dark age'. India was also a comparative backwater by this standard, completely lacking in unity, historiography, or a unified cultural tradition - archaeology is, realistically speaking, our only light on the subcontinent until the second millennium CE, and despite the large population of India at the time, it was in no position to impose anything on anyone. Only China can compete with or exceed Europe's historiographic tradition and cultural continuity over the last 2,500 years. The idea that Europe was so backward and everywhere else in Eurasia so advanced is predicated on a double standard.
I mention this because it is repeated in Jason Antrosio's latest piece on Diamond's theories about the rise of Europe. In fact, he goes as far as to say that in the ninth century CE,
One can imagine yet another perplexed European Yali, seeing a bustling Near Eastern city for the first time after being sold into slavery from a forest hinterland!Yali being Diamond's interrogator at the beginning of Guns, Germs, and Steel, the one who asks him why it is that white people have so much cargo.
You ought to be able to see why this is wrong: Yali comes from New Guinea, an island of horticulturalists who live(d) in villages and had no tradition of metallurgy. Europeans in the ninth century had almost all of the technological capabilities of the Arab caliphates, including metallurgical traditions, and many of them lived in large towns. Little about the experience of living in a city would be foreign to a Byzantine or even a Viking - not because all Europeans lived in cities, but rather because cities formed part of the backdrop to their lives, as they do for anybody who lives near a city. Cities and large towns were common in Europe at the time, and had been for centuries. Life in 800 CE Rome and Damascus would have been pretty comparable: you would have died young, surrounded by diseases, trans-continental trade, gold finery, and the products of empires, kingdoms, and non-state societies from across Eurasia. Likewise life in Winchester or Paris or Cordoba or Constantinople.
That isn't to say that European domination of the earth would have been obvious in 800 CE. But it's not as if the Europe of the fifteenth-century CE was a miracle of urbanisation and literacy that appeared on the scene without any antecedents in the so-called 'dark ages'.
*I'm reading Christopher Ehret's An African Classical Age at the moment. My comments aren't intended to denigrate Africa, but rather to show that we tend to have higher standards for Europe than elsewhere. Everyone seems to think that Europeans were uniquely smelly, backward, stupid, unclean, and ugly in the early Middle Ages, and that everyone else was clean and lovely and had excellent sanitation, but this isn't true.