The impact of European exploration on American civilizations cannot be understated. In fact, the history of American civilization can broadly be divided into events prior to 1492 and Columbus's voyage to the New World, and everything that has happened since.
European powers quickly realized the value of colonizing the Americas, taking over regions in order to gain wealth like gold and silver, establish trade networks, and find new people to (often forcibly) convert to Christianity. This often required the destruction and subjugation of native civilizations: the Spanish, for example, quickly conquered the two largest American empires, the Aztec and the Inca, in order to establish their own authority and force the natives to work on their behalf.
One of the greatest impacts of European exploration was the spread of diseases. Over the course of hundreds of generations, Europeans had become highly resistant to diseases like smallpox, measles, and fevers. Upon landing in the New World, these diseases spread quickly to the native population, who had no such resistance to speak of. An estimated 90% of the population perished, making it easy for Europeans to take over whoever was left and to expand into these newly empty areas. Population estimates of the pre-Columbian Americas may be as high as 100 million people; just a century or two later, it is doubtful even half this many people were living in both continents, due to epidemics and violence waged by European invaders.