What impact did European colonies have on Native Americans? How did the Spanish establish an empire in the Americas?

This is a very broad question.  In general, European colonization had many impacts on Native Americans.  In some cases, these were positive such as the trade relationships that were opened between the French and some Native populations in what is now Canada. In some cases, they were neutral, such as intermarriage and the spread of new cultural ideas in both directions. In some cases, they were negative, bringing epidemic diseases that wiped out Native populations and violence that forced Native populations to relocate or be overtaken by European colonizers. 

There were several factors involved in creating the Spanish empire.  One was gunpowder.  Since many of the Spanish conquerors had guns and by and large the Native populations did not, guns were not only a deadly weapon but also a factor in intimidating and psychologically overpowering populations.  Horses were another factor.  Many of the Spanish warriors were mounted on horseback.  This was both a military advantage and also a tool for manipulation, as some of the Native populations, most notably those in what is now Mexico, had legends that featured heroes or gods on horseback, so there was another tool for psychological conquest. In addition, epidemic diseases also played a role, since diseases that were common in Europe were unheard of in the New World and easily decimated populations.  Once these conquering factors had overwhelmed civilizations, as they did in many places, the tools of religious teaching/conversion and civil government (i.e. taxes and laws) were used to begin to build an empire in the Americas.