This is a great and unfortunately very complex question, for which there are a number of perspectives. While the issue is still debated, in recent years something of a consensus has emerged among economic historians: the discovery of overseas trade routes, acquisition of colonies, and creation of a stable banking sector were the three developments that triggered Western Europe's "Great Divergence" in wealth and military power compared to the rest of the world.

First, though, we have to realize that Western European "domination" was never universal, nor simple. There are a few caveats we have to keep in mind:

1. The process of Western Europe's divergence was incredibly slow, even at its "fastest" in the 19th century. The average growth rate of the British economy at the height of the industrial revolution barely exceeded 1%, which is excruciatingly slow by modern standards.
2. The rise of Western European countries had just as much to do with the decline of non-Western countries than their own institutions and reforms. China, for example, remained the world's largest economy until the mid-19th century, but already before then was losing wars simply because of the internal weakness of the Qing regime. This is part of the reason why Western Europe's seafaring capabilities were so critical to its dominance - Western Europeans could pick and choose who to attack and when.
3. Going off of this, some countries were never dominated by Europeans for any extensive period of time. Japan is the prime example, which on two occasions, centuries apart, rapidly modernized its military to meet and even exceed Western European standards.

With those in mind, we can start to build a clear picture of Western Europe's rise to outsized geopolitical importance from the 15th to 19th century.

Exploration and the Birth of Seaborne Mobility

As mentioned in point 2 above, Western European imperialism, especially in its earliest phases, depended on opportunism which was enabled by the ability to quickly transport armies over long distances by sea. This was fuelled by a rapid process of naval innovation which started in the early 15th century in response to an increasing focus by Spain and Portugal on exploration and cross-ocean activities.

In 1415, Dom Infante Henry the Navigator played a key role in taking the Moroccan city of Ceuta in North Africa. In 1419, the Portuguese discovered and colonized the Atlantic island of Madeira, on which they planted sugarcane, which was then controlled by Venetian traders. These events made further exploration down the African coast seem like a potentially lucrative opportunity, and triggered a century of prolific discovery sponsored by the Portuguese government. Already in 1474, Portugal had created a colony on the African mainland, and in 1478, the Spanish tried and failed to take it in Europe's first ever colonial war. 10 years after that, the Portuguese rounded the Cape of Good Hope, and in 1492, Christopher Columbus made landfall in the Americas. Just six years later, Vasco de Gama arrived in India.

As the Portuguese built trading outposts along their route to India and the Spanish heard ever more enticing tales of the riches of "los Indias", both countries entered what was probably the first blue water naval arms race. During this period, quantities of armed, oceangoing ships and seafaring technologies advanced. In 1451, the Portuguese developed the fast and mobile caravel, based on designs in use in Iberia since the Muslim period. This and other naval innovations allowed the Portuguese to win several naval battles against non-European powers, the largest of which was the Battle of Diu.

The advent of European "Seaborne Mobility" enabled an opportunistic policy by the Portuguese. Throughout the 16th century, Portugal tried to pursue good relations with strong countries like China, Siam, and Vijaynagar, while attacking weaker countries like Johore and Gujarat. Portugal was able to acquire trade "outposts" (in many cases entire cities) through this policy and secure favorable trade deals which were a forerunner to the 19th century unequal treaties. In these arrangements, Portugal was often the exclusive European partner of the African or Asian party, enjoyed low or nonexistent customs, and exempted its subjects from local law.

Terrestrial Empire

Spain, meanwhile, had acquired an impressive "terrestrial empire" by using its first rate military doctrine to acquire the Americas' major population centers. Spanish military traditions stemmed from the days of the Almogavars - proto-guerilla fighters that raided Muslim-ruled territories from the 10th to 15th centuries. Using highly mobile tactics, marching long distances, and understanding war as an economic proposition (with leaders prioritizing loot to keep subordinates loyal), the Almogavars were called Europe's "finest infantry" at varying points during their existence. By the time the Spanish knighthood became a closed caste in the 12th-13th centuries, many former raiders had entered the class. The Spanish nobility, which by the colonial era were called hidalgos, took to the field more often than their counterparts elsewhere in Western Europe (who had largely ceased to be a "warrior" nobility by the 13th-14th centuries), and inherited a cunning military ethos (and contempt of productive work) from their forefathers.

As a result of their innovative tactics, adeptness at traversing long distances in harsh terrain, ability to manage the morale of their soldiers, and technological edge, Spanish hidalgos were able to conquer the Americas' premier empires with ridiculously small armies. They leaned heavily on native allies to do this and prioritized loot. As a result, they were able to win at minimal cost to the Spanish crown (though, as a caveat, they were also often defeated).

In conquering the Americas, conquistadors divided up the land into Encomeindas - estates which were closely modeled after the Spanish feudal system. People living on these lands had a status roughly analogous to Spanish peasants, but with far fewer legal protections. Due to much of the population dying from disease and death, Spanish-American colonies imported large numbers of slaves. With extensive bullion reserves and climates often suited to growing some of Europe's most prized crops, Spanish-American colonies increased the revenue of the Spanish state, leading it to become arguably the world's premier military power by the mid-16th century.

This is one of the very frequently asked questions here. See the thread /u/profrhodes wrote here. Now, this is by no means an uncontroversial topic and there is certainly no absolute concensus about it but I think his posts covers the mainstream view pretty well.