If you want to be able to make your permanent home in the United States, you'll need what is called permanent residence, or a green card. Green card holders can live and work in the United States and travel in and out, with very few restrictions (though they can't vote and can be deported if they abuse their status or break U.S. laws).
Family members of U.S. citizens make up the largest number of green cards issued each year. Others are issued to investors and workers who have been petitioned by U.S. employers or have special skills. Still other categories have a humanitarian basis, such as refugee or asylum status (which can lead to a green card), for people who are fleeing persecution.
People who want to come to the United States for a limited time need what is called a nonimmigrant visa. This lets them participate in specified activities (such as studying, visiting, or working) until their visa runs out. Students and businesspeople make up the largest groups of nonimmigrant visa holders. Nonimmigrant visas are also issued for tourists, exchange visitors, and workers with some kind of specialty that is lacking in the U.S. workforce.