Another possible scenario is when implementing an embedded processor in an FPGA device -- a soft processor, you have to configure the FPGA device to behave like a processor system using the default configuration bitstream and after that you can boot up the processor system. However, reconfiguration during run-time is not possible without external help, for example you may have a controller which is responsible to reconfigure the FPGA device. In this case the process will put itself in a safe state (e.g. saving its data in an external RAM, etc.). It then notifies the configuration controler to reconfigure the FPGA device with a new configuration. After reconfiguration, a new processor becomes active.
Since some FPGAs support partial reconfiguration, it maybe possible to reconfigure only some parts in the FPGA. For example, a function unit in the datapath of the processor can be replaced with another function unit after reconfiguration.