If you are a J-1 visa holder admitted to an academic program in the United States, 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 make this switch, one of the most important issues will be whether you can provide proof of liquid assets sufficient to pay for one full year of your school tuition, living expenses, and insurance. You will also need to be able to cover expenses for your dependents, if any (spouse, children, or other people whom you support).
Your academic institution will be able to provide you with estimates for many of 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. This requirement is common to all nonimmigrant (temporary) visas, though it can be challenging for students who might not have a job or home awaiting them in their country of origin.
Many J-1 visa holders are subject to something called the two-year home country physical presence requirement, which mandates that they spend two years in their home country before returning to the U.S. on any of most other types of visas (including immigrant visas, which are given to new permanent residents).
The good news is, J-1s subject to the two-year residency requirement are still eligible to change their status to F-1. But it won't make the requirement go away altogether. It will have to be fulfilled at a later time, assuming the change of status is successful and the person later wants a different type of U.S. visa or status.
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 (the "Certificate of Eligibility for Nonimmigrant Student Status"), you can apply for F-1 status. You have a choice of doing so by either 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 procedure has its advantages and drawbacks. Consult an experienced immigration attorney if you have questions or concerns.
For a more thorough discussion of the change of status (COS) process, see Applying for an Extension of a U.S. Visa or Change of Status.