The impact of the meeting between these two groups of peoples was much greater on the Native Americans than on the Europeans. However, both groups were affected.
For the natives of the Americas, contact with Europeans changed their entire way of life, resulted in the loss of their autonomy, and almost annihilated their population, completely eliminating it in some parts of the two continents. The Europeans brought technologies (and horses) that completely changed Native American ways of life. They used some of these technologies to dominate the Native Americans. In North America, they tended to push the natives off their ancestral lands while in South America they subjugated the natives. Either way, the natives lost their autonomy. European aggression, combined with European diseases to which the natives had no immunity, destroyed as much as 90% of the original population of the Americas. These were huge impacts that can be clearly seen.
The Europeans were not impacted in such obvious ways. However, coming to the New World did impact them as well. Contact with the New World changed the politics of the Old World. It enriched Spain, making it a greater power. Contact with the indigenous people created a large group of mestizos in Latin America, creating a new, hybrid society. Contact with Native Americans can be credited with teaching English and other settlers in North America some of the skills that they needed to expand out into the wilderness (ironically, pushing the Indians farther into marginal lands). In these less obvious ways, contact also had an impact on the Europeans.
It's easy to enumerate the effects on the native population -- diseases must be mentioned, and should the introduction of firearms, as well as the complete loss of lifestyle. But the effects of the native American on the European population were subtler -- Christians, for example, were introduced to the idea of Nature being worshiped and acknowledged as a universal "truth," an idea lost to Europe since the Druids. Also, the concept of "owning land", taken for granted since the habit of kings awarding large land tracts to loyal soldiers and defenders, was challenged by the Native Americans' attitude that land was by definition un-ownable. Finally, the idea of expansion itself, though fairly alien to the natives, changed the Eruopeans because there was no "virgin" unclaimed land left in Europe.