Visa vs. Status
A visa is a physical stamp inside an individual's passport issued by U.S. Department of State at a visa-issuing post abroad. Immigration status is a classification with a set of rules that govern the individual who has been admitted into the United States under a particular category. For example, a J-1 student visa will usually allow an individual to be admitted into the U.S. in J-1 student status. Once an individual been admitted into the U.S., the physical visa is of no intrinsic value until s/he leaves and want to reenter the United States. However, immigration status remains critically important. For this reason, the actual visa in your passport can expire while you are in the U.S. but you must maintain legal status throughout your stay. When an individual is legally admitted to the country, they acquire "STATUS". This term refers to the legal presence within the country, and the U.S. Department of Homeland Security regulates it. Just like visas, there are many categories of status, which are defined by non-immigrant classification designations. The I-94, received at the time of entry, will indicate the designated status as well as the expiration date of that status.
A student in F-1 or J-1 status may legally remain in the country for "Duration of Status", which will be indicated as "D/S"on their I-94. This means that as long as the person is doing what the regulations specify they need to do to maintain their F-1 or J-1 status, then they have status until they are done with their specified academic program, plus an additional grace period (60-days for F-1; 30 days for J-1) to prepare for departure from the country. Other status types, such as B-2 (tourist) or H-4 (employee dependent), will have a specific expiration date for their status, and they must depart the country by the specified date or face possible legal consequences, including potential deportation and/or being barred from future admittance to the U.S.
For more information about visas, please visit the U.S. Department of State's website.