European explorers, initially from Spain and Portugal but increasingly from France and Great Britain as well, began to colonize the globe well into the beginning of the seventeenth century. Columbus' voyages merely solidified the reality of Europe's growing transnational hegemony. Vasco de Gama, for example, had discovered a connecting sea route to India in 1499, and since that time European monarchs had been interested in finding a westward passage to the valuable spices of the East Indies. Columbus's discovery of the New World in many ways just represented a culmination of expansion that had begun with the voyages of explorers like de Gama and others.
Discovery of the New World provided European monarchies with an enormous amount of resources with which they could grow their overseas empires while engaging in intensive industrialization at home. The Spanish, for example, initially made colonies in St. Augustine in present-day Florida, Central America and Peru. After ravaging the local peoples in these regions, Spanish conquistadors stole massive amounts of silver, which they used to enrich their home governments and themselves. The influx of this silver drew more and more Europeans over the Atlantic in search of the kinds of riches the Spanish had come upon, and to meet the growing demand, European monarchs made out several charters for the establishment of New World colonies and joint-stock companies.
When Spanish influence in the New World started to decline in the mid-seventeenth century, the British took over as the primary settlers. Their focus on a plantation-based economy, particularly the cultivation of crops like sugar, tobacco, and cotton, produced enormous wealth, enabling the British crown to fund more voyages, send more supplies, and engage in more extensive warfare with the indigenous inhabitants. Furthermore, the massive wealth generated by the sugar plantations in particular allowed the British to expand their empire elsewhere.
By the beginning of the 1800s, the British had holdings in West Africa, Australia and New Zealand, India, much of North America, Guiana, and elsewhere. Similar material windfalls derived from New-World profits allowed the French, Dutch, and Germans to expand across much of the world as well.
Ultimately, local peoples in the places that Europe came into contact with could not defend themselves against incursion. By the 1800s, European empires could draw on resources from all over the globe and could almost endlessly reinforce colonial outposts that they deemed worth maintaining with men and material. Europe was essentially able to take over the world in large part because of the wealth and power that these early successes in the New World provided.