The situation varies in the United States. Typically, undergraduate students have a choice.
1) They can choose to live in a university dormitory ("dorm.") Housing is not included in the price of tuition. It is a separate fee.
2) In some universities, they can choose to live in a fraternity or sorority. These are complete private group living units. They are organized as clubs and you have to be invited to join one. They have to like you and think you fit in with their group's culture. It is very, very common for these organizations to have names consisting of two or three Greek letters: Delta Kappa Epsilon, Sigma Alpha Epsilon, Delta Gamma, and so forth, and they are often known collectively as "the Greeks."
3) Undergraduates can live in private housing. This can be an ordinary apartment, but apartments tend to be much too expensive for students, and renting arrangements don't fit in well with the school year. In college towns, a variety of housing options tend to spring up. Houses will be converted into apartments; people build apartments intended to be lived in by, say, four students (with a single bathroom and kitchen); or people may actually build "private dormitories."
It's very common for universities to require that students live in university dormitories for their first two years (freshman and sophomore).
University dorms are almost always the cheapest alternative. When I was in college (in the 1960s) college dorms tended to be somewhat shabby and lacking in luxuries, but competition, particularly among private universities, has resulted in modern dorms being nicer and more luxurious than they were.
I would strongly recommend that students from outside the U.S. begin with a university dormitory. They can look around for other possibilities during their first year.