In all the worst ways. Throughout history, indigenous people have been conquered, abused, enslaved, even murdered by foreign invaders. This is not a uniquely European behavior; most cultures throughout history have engaged in some form of imperialism against other cultures. However, European technological, economic, and above all military dominance starting around the 15th century led to a global rush to colonize and establish empires around the world, which shaped world geography as we know it. North and South America were thoroughly colonized by European powers, as was Africa and much of central and south Asia. The only major exceptions were in East Asia, where China and Japan had powerful civilizations of their own and largely avoided being colonized (and in some cases colonized other people).
Some countries did relatively well, eventually: The United States and Canada are both British colonies, and are quite prosperous today. Others did not: Ghana and Cote D'Ivoire were also colonized, and remain in abject poverty today. Even in relatively prosperous countries such as the US, indigenous populations suffered substantial violence and discrimination for centuries. Even today, indigenous people such as Navajo and Inuit are not always treated with equal respect by the European-descended majority; though at least in terms of law they formally have equal rights.
In addition, the transfer of foreign diseases frequently destroyed populations which had no evolved immunity; the smallpox pandemic in North America was completely catastrophic, worse even than the Black Death had been. This was actually the main cause of death of most indigenous populations, with imperialist violence being second.