In a computer error message, a fatal exception indicates an exceptional situation requiring that the program responsible for the situation be closed.
In general, an exception is any uncontemplated situation (which includes but is not limited to program errors). A fatal exception simply means the exception can't really be handled so that the program can continue to run.
Software applications communicate with the operating system and other applications through several different layers of code. When an exception occurs at a certain layer of code, each layer sends the exception to the next layer in order to locate any exception handling code that can deal with the specific exception. If no exception handling code exists (at any layer) to deal with the exception, a fatal exception error message is generated by the operating system. This message can also contain some cryptic information about where the fatal exception error occurred (such as the hexadecimal location within the program's range of memory). This extra information has little value to the user, but may help support people or developers in debugging an application.
When a fatal exception occurs, the operating system has no other recourse but to shut down the application, and in some cases, its own operating system. When using a specific application, if you continue to experience fatal exception errors, report the problem to the software vendor.