As a general rule, if you hold lawful non-immigrant status in the United States you have the opportunity to file to change to another visa status in the United States. This means, for example, that you may have come to the United States as a B-2 visitor but, if after your arrival you decide to study in the United States and you have been accepted into a college, you can file to change from a B-2 status to an F-1 student status. The most important factor to keep in mind if you wish to remain in the United States is to file your application for change of status before your status expires.
While most people lawfully in status in the United States have the opportunity to change their status, the main exceptions are for people who are here on visa waiver status, who may not either change or extend their status, and for people who are in J-1 status and subject to a two year foreign residency requirement. Many people on J-1 are not subject to the two year foreign residency requirement, and even those who are subject may still change to certain visa statuses. The moral of the story is that it is always wise to check with an immigration attorney if you are on a J-1 and thinking of changing your status.
There are multiple issues involved in the process of changing your visa status and the process can be more complicated than it appears; it is advisable to consult with one of the attorneys at Landau, Hess, Simon, Choi & Doebley before engaging in this process.