To study in the United States, you'll need a student visa, which depends on the type of study you're doing:

F-1 visa
The most common visa for academic study at an accredited college or university. You can apply for an F-1 visa if you've been accepted to a SEVP (Student and Exchange Visitor Program) certified institution. You can use an F-1 visa to study at any course at the institution, and you can travel freely within the US and abroad while your visa is valid. However, you must prove you intend to leave the US when your visa expires.

M-1 visa
For vocational study at a recognized nonacademic institution, such as a business school or post secondary vocational school.

J-1 visa
For students participating in an exchange visitor program, or for those taking up prearranged employment, training, or research in the US
You can apply for a visa once you've been accepted to a course, and it can take 3–5 months to get a visa after you apply. However, visas can be issued up to 120 days before your departure.
Even if you get a visa, Customs and Border Protection (CBP) has the final say on whether you can enter the US and how long you can stay. When you arrive, a CBP official will validate your Form I-94, Record of Arrival-Departure, which will note your permitted length of stay.
If you want to stay in the US after your degree, you can re-enroll in a higher program or change your visa status.