Of course the answer to this question regards only the time after the 15th century. The European dominance started certainly only with the age of discoveries. Before this time the European culture group was not particularly more advanced than say the American High cultures or those in India, the Middle East and China.

China for example had an enormous fleet of exploration ships under the eunuch Admiral Zheng He, he sailed along the coasts around India as far as Arabia and the Horn of Africa. That was 60 years before Columbus. What he didn't have was the knowledge how to navigate far away from a coastline, and even more important when a new emperor ascended the throne China chose the way of isolation for several centuries (they had a painful awakening in the late 19th century, when Europe had already divided the world).

The Muslim world was of course very advanced in science, astronomy, medicine, philosophy - and yes also seafaring (it's not a coincidence that the Caravella ship type came from the Iberian Peninsula, where Muslim and Christian realms co-existed for centuries.

But looking at Europe now: It is of course one of the most diverse continents. There's not truly a great landmass in comparison to the vastness of Asia, Africa or the Americas. Different cultures evolved in a diverse climatic environment: ideas, goods, and innovations could be exchanged relatively easy (mountains might be a natural border, but they are easy compared to a desert).

The different cultures and realms of course stood in competition to each other, but their religion was a uniting factor. Without a doubt the missionary aspect of colonization should not be underestimated. There was even legends that ignited the wish to explore and find other christian societies (check the legend of Prester John: his Christian realm was believed to be found at times in India, Mongolia, Ethiopia, Georgia, Persia, Central Africa).

When the Europeans eventually discovered America - searching for trade routes for luxury goods, the silk road was controlled by the Muslims - they had some advantages over the native population, among them such highly developed cultures as Inca, Aztecs, Maya. They were mostly of military nature: Canons, Horses, Steel - as you might remember Hernán Cortés conquered the giant Aztec empire with a few horsemen and a handful of canons. That doesn't mean the Europeans were superior in any way, only the advantages of the American civilizations didn't help them in their situation.

But more important than such technological gimmicks was of course biological coincidence: As I said above the diverse geography of Europe allowed diverse cultures to develop - and also helped diverse bacteria and viruses to evolve (with a equivalent reaction of people's immune system). The diseases the Europeans were immune to were new to the American peoples - they literally died like flies. And if a European was infected with something, he died long before he could carry his sickness back to his continent.

Ok, so by this point the Europeans had free range in the Americas: they were really creepy strangers bringing death and destruction so you give them what they wanted - they took it even after you agreed with them. That was gold, and that stuff really filled the coffers of the European powers: Portugese and Spanish (also the English who developed a habit of pirating Spanish galleons).

And when money gets involved people get really inventive to get more: the Europeans developed anything that could allow them to cross and explore the oceans for more wealth. And they developed the means to defend their riches.

So that's about the beginning of the European dominance, where all the foundations were laid for the next four-and-a-half centuries