This question is at least a little difficult to answer because the US “dominated” different countries to different extents. The US had a relatively small empire in terms of places that it actually took formal control over. However, there were many more places over which it exerted various degrees of control.
There were only a very few places that came under direct and total control of the United States. These places included American Samoa, Hawaii, Puerto Rico, Guam, and the Philippines. However, the US took some amount of control over many other countries. Most of these countries were in the Americas. On the lower end of the scale, it exerted a great deal of economic control over Mexico. The US did not do much in the way of actually sending troops into Mexico (at least after the Mexican-American War), but it did use its economic power to have some control over that country. On the higher end of the scale, the US sent troops into such places as the Dominican Republic, Nicaragua, and Haiti. It did so largely in order to protect American economic interests and to keep other countries from sending their own troops in. Finally, there was the case of Cuba. The US exercised veto power over that countries affairs even as it allowed the country to be nominally independent.
The US did these things for three main reasons. First, it wanted an empire for economic reasons. It wanted sources of cheap raw materials and it wanted markets for its own goods. Second, it wanted an empire for military reasons. Having an empire allowed the US to project its power around the globe. For example, having the Philippines gave the US naval bases with which to project power in Asia. Finally, it wanted an empire for reasons of prestige. It wanted to have the prestige that went with having a big empire.