This thread is too long to comment on in its entirety. There are some really smart comments here (many about geography) and some... Er... poorly informed ones.
Here’s the gist of what I tell my college students:
1. The Jarod Diamond thesis is reductive, but not entirely wrong (Alfred Crosby did it first and better, so skip Diamond). For instance, microbes matter, especially in the history of the Americas post 1492. However, our current understanding of the major histocompatibility complex suggests that diseases decimated indigenous populations because they were accompanied by food shortages, social upheavals, and related changes caused by European conquest. So, it kind of is a guns, germs, steel story. Of course strong indigenous resistance lasted a lot later than most kids are taught in school.
2. China and India are key. Those two countries accounted for two-thirds of global GDP as late as 1800. By the mid-twentieth century, we called those nations part of the “third world.” So, why the rapid change in fortunes? For India it has a lot to do with the textile industry (cotton) and Britain’s systematic deindustrialization of India to transform it into a raw material producer (after the start of crown rule in the middle of the 19th c.). For China, the answer lies in the Opium Wars and free trade imperialism. Opening up China for trade with the rest of the world was a longstanding goal of Western Europe. The Opium Wars provided that opportunity.
3. From backwater to powerhouse. The rise of Europe can be attributed to a lot of factors. The demographic catastrophe of the Black Death spelled the beginning of the end for feudalism and a slow emergence of a new middle class. Long distance navigation (as someone else mentioned) was also key. Private property and capital accumulation also helped to facilitate the emergence of the modern fiscal-military state (think England during the Glorious Revolution and the Enclosure movement). European warfare (using Chinese inventions) was also brutal and probably helped sharpen the war-making skills of European countries. The wealth of the New World enriched many European nations. New World commodities like sugar were lucrative. Quinine was pretty transformative (since it is a fairly effective anti-malarial). It originated in Peru, but by the late 19th c. was being grown in the East Indies and consumed with gin as Europeans chugged along in steamships up rivers in sub-Saharan Africa to brutally claim territory. Lastly, England, in particular, was lucky enough to have easy access to coal so that it industrialized quickly and overcame the biological limits imposed by its denuded island landscape. Nations like China weren’t so lucky.
Geographical luck and colonialism is the shortest answer, but that’s clearly just scratching the surface.