There were many clashes between the Native Americans and early European settlers; though much of it ended up in "mezcla" - a mix. For example, Europeans brought their Christian/Catholic beliefs, and as a result of European attempts to convert Natives to their beliefs, Natives (with the help of missionaries) combined many Catholic beliefs with their own, turning their "gods" into Catholic Saints. This, actually, is where we get many of the Saints that you'll hear about in Mexico and South America that Northern Catholics aren't familiar with (Our Lady of Guadalupe, for example).
Property ownership was different, as well. In Native American communities, an individual rarely owned something, especially land. They more so owned the right to live on that land, but it didn't mean the rest of the community had to stay away. So, if a Native sold, for example, some riverfront property to a settler, the settler might be shocked to see several people fishing off his lawn the next day. The Natives didn't understand the problem - how could anyone own a piece of nature? The Europeans, on the other hand, felt Natives had improperly taken back what they'd sold and gone back on their word - hence the nasty term "Indian Giver."
Another difference was the matriarchal versus patriarchal households and communities. Europeans normally had men in charge of families and communities, whereas Natives often had women running the show.
Basically, settlers (and later Americans) did what they could to "civilize" the Natives, trying to convert their "heathen" ways to European standards. This, of course, led to much conflict, unfortunately with Natives often losing their lives or being evicted from their own lands.