FPGAs have a bunch of logic cells that need to be initialized with a stream of configuration bits. This bitstream usually comes from a flash chip located outside the device, although some devices have flash memory on board.
Partial reconfiguration means that only some logical cells can be configured, while the rest are used. This is typical for specific models.
A complete reconfiguration is possible, even if your device does not support it - you will need to reprogram the flash chip, and then execute the Reset command or reboot.
Some devices have more than one configuration image in the flash configuration. The device will download the first image, and if he does not like it, it will download the second (or subsequent) image. This may be for redundancy or sets of distinguishing features.
Some of the FPGA SOCs (e.g. Xilinx Zynq) use the microprocessor core to load FPGAs. In this case, the microprocessor core can change the FPGA as much as is required during operation.
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