What we call the Rocky Mountains is really two different mountain chains, one on top of the other. The older mountain chain started billions of years ago, before there were any plants or animals living on land. The only living things on Earth at that time were one-celled creatures floating around in the oceans.
Most of the rocks of these younger mountains are igneous rocks like lava that shot out of volcanoes when the Earth was first forming, or metamorphic rocks like gneiss made out of that igneous rock. These rocks got pushed up into mountains because of plate tectonics, when the Pacific plate with California, Oregon, and Washington on it began pushing against the plate that had the rest of North America on it. Because of how the plates were shaped, the Rocky Mountains formed not on the very edge of the plate, but several hundred miles further away. These mountains stretch all the way from the southern United States up into Canada.
Like the Himalayas, the Rocky Mountains are still growing even today as the Pacific plate continues to push against North America.