One of the first major consequences had to do with the arrival of European diseases which ravaged their civilizations. Things like smallpox, to which the Native Americans had not been exposed, killed millions of them, which also served to weaken many of their societies and left them open to exploitation and invasion that might have been less successful had they not been so weakened.
And the Europeans were obviously not just interested in spreading disease (in fact most of that happened without their intention), they also looked to take control of the land, the mineral resources, and to convert the heathens (in their eyes) to the proper religion.
Over time, the Native Americans went from being the dominant people on the continent to being forced into subjugation or even simply expelled from their lands as was the case in most of North America. Their interests and rights were almost always considered to be less important and worth less than those of the incoming European explorers and then settlers.
There were quite a few consequences that the Native Americans suffered due to Europeans. When Europeans first moved to the Americas, they brought many illnesses that the Natives had not experienced before. Smallpox is the best known example. In Europe, illnesses began because of their living situations. Too many people were put into one area and it caused dirty and unfit conditions. Illnesses spread very quickly and if you survived to become an adult, your immune system was pretty good. Many people did not survive the many childhood illnesses that affected Europe. The Native immune system would not have experienced any of these illnesses due to their mobile lifestyle. When Europeans came, those illnesses were given to the Natives and many of them died because of this. Some illnesses directly targeted people in the middle years of their life. This caused entire tribes to lose a generation of healthy, young men and women. This not only affected how they functioned as a society, it also affected the next few generations.
The Natives were not looked at very highly by the Europeans. The Natives moved from place to place based on the seasons, gave their women high status in their society, bathed regularly, and had a great respect for nature. All of these concepts were foreign to Europeans and so they thought of the Natives as savages and barbarians. They believed they must be taught to be civilized and assimilated into the European society.
This problem was only furthered when it came time to negotiate for land. The Native Americans did not have the same ideas on ownership and value as the Europeans. The Natives believed the land belonged to no one and it was only there to take what was needed. The Europeans wanted to use the land for farming and building cities. Many Native tribes lost their lands because the Europeans took advantage of their lack of understanding. An example: a Native tribe agreed that the Europeans could use a "buffalo hide" worth of land in the area they traveled to in the summer. To the Natives, this was not a large piece of land. They left for the fall and winter seasons and returned the next spring to find a city had been built on the lands they had used. When the Native leaders asked the Europeans why they used so much land, their response was a "buffalo's hide" could be cut and stretched in order to get more material out of it. They used this excuse to take over a much larger piece of land then what had been previously discussed. The Native then had a find a new place to camp for the summer as the land they had used for generations had now been taken over by Europeans.
When the Natives began to realize that the Europeans were tricking them, they no longer agreed to land treaties. When legal options could not get Europeans land, they would take it by force. The army would be deployed in an area that the Europeans wanted for their own. Entire tribes lost their homes, many Natives were imprisoned, and some were shot if they refused to leave. This left Native groups even smaller than before.
Eventually, the Native groups had been so decimated by illness and war that they were no longer a threat to European expansion. The Natives not only lost their homes, but their families, culture and way of life.