Clarifying some points that @indigochild doesn't address.
Any lawful permanent resident who has been in the United States with lawful permanent resident status long enough, who completes the application and passes certain tests, and is not disqualified by factors that don't include nationality can become a U.S. citizen.
Also, often the naturalization of an adult will automatically lead to the naturalization of their minor children without the same formal requirements for eligibility. Minors who become citizens in this way don't generally have to take an oath and are an important source of dual-citizens. (Another common path to dual citizenship involves having U.S. citizenship by virtue of being born in the United States, and non-U.S. citizenship by virtue of having a parent who is not a U.S. citizen when you are born.)