To visit the U.S. you must obtain the right type of U.S. visa. You will not be able to enter the U.S. without a visa. If you just want to visit the country and not settle there, you must get a visitor visa, also known as non-immigrant visa.
To travel to the U.S. for business or for pleasure, you can get B non-immigrant visas, also known as visitor visas. If you have to enter the U.S. for reasons other than business or pleasure, you must apply for the right type of non-immigrant visa. There are different types of non-immigrant visas like work visas, student visas and fiancé(e) visas. Depending on the purpose of your trip, apply for the right visa. For example, if you seek to study here, you can apply for F or M student visas.
A non-immigrant visa will permit you to stay here only for a certain period of time. The date on which you must depart the U.S. will be stamped on your passport or you can find it on your Form I-94, Arrival/Departure Record, issued by U.S. Customs and Border Protection (CBP), at the U.S. post of entry. You must not stay back in the U.S. after that date as you will become deportable if you do so.
In case you wish to remain in the U.S. for few more months beyond your projected date of departure, you can request an extension of stay by filing Form I-539, Application to Extend/Change Non-immigrant Status, with U.S. Citizenship and Immigration Services (USCIS). If your request for an extension of stay is approved by USCIS, you will be allowed to stay in the country beyond the time indicated on your passport.