The Spanish went to North America with a mind to conquer (oh, and also to steal gold). There were many Spanish attempts to take land, whether through combat, converting the natives, or through using sickness to their advantage.
Spain used conquistadors, which were professional soldiers to fight the natives in their lands. The conquistadors used European weapons, such as matchlock guns to fight the natives.
Many people went to North America to convert the Native Americans to Christianity. These missionaries established missions, which were places where they taught the Native Americans Christianity, Spanish customs, and many other Spanish traditions. Some of these Native Americans became Spanish citizens, however, when the missions sought to gain control over the Native Americans, many of them revolted, attacking the missions and destroying many of them.
Finally, the Spanish used European sicknesses to wipe out most of the Native American population. In fact, one of the first examples of biological warfare was the siege of Tenochtitlan, where Hernán Cortés fought the Aztecs using European weapons, horses, enemies of the Aztecs, and the smallpox disease, which killed much of the nations' leaders and thousands of other people, though the Spanish were not affected because of their immunity to the disease.