If you are a J-1 visa holder admitted to an academic program, you are permitted to change your status to F-1 if you intend to pursue studies as a full-time, academic student.
In order to do this, you must provide proof of liquid assets that are sufficient to pay for one full year of your tuition, living expenses, insurance, and expenses for your dependents (if you have a spouse, children, or other dependents). Your academic institution will be able to provide you with estimates for these amounts.
(See Financial Requirements for a Student Visa for more detail on this important eligibility issue.)
As an applicant for an F-1 visa, you will also have to provide proof that you intend to return home at the completion of the academic program. For more information on how to prove this, see How To Apply For a Student Visa.
J-1 visa holders subject to the two-year foreign residence requirement are eligible to change their status to F-1. It is important to remember that the two-year foreign residence requirement will still have to be fulfilled if the change of status is successful. If you are subject to this requirement, see How to "Waive" The J-1 Two-Year Home Residency Requirement for information on how to have it lifted in your case.
Once the academic institution that accepts you as an academic student issues you an I-20 form, you can apply to change status either by traveling home and applying for the F-1 visa at your local U.S. consulate, or by submitting an I-539 Application to Extend/Change Nonimmigrant Status to U.S. Citizenship and Immigration Services (USCIS). Each process has its advantages and drawbacks and you should consult an immigration attorney if you have questions or concerns.
For a more thorough discussion on the change of status (COS) process, see Applying for an Extension of a U.S. Visa or Change of Status.
In any instance when you want to change your visa status, you should strongly consider speaking with a US immigration lawyer - prior to making any application to USCIS - in order to understand the application requirements for the visa you are trying to obtain.