[zwieback]

Probably burned into an external EEPROM or flash, the FPGA fabric itself still has to be configured by some external means. It's pretty fast but on-the-fly reprogramming of the FPGA logic is still an issue where you want to very quickly switch the logic on the FPGA.

[monocasa]

Most FPGAs I've seen go out of their way to allow you to reconfigure while it's running, you just have to jump through a few extra hoops and get down and dirty with the floor plan.

[kevin_thibedeau]

Older Altera CPLDs and Actel/Microsemi antifuse based FPGA families have nonvolatile configuration within the fabric. They enjoy the benefit of near instant start up since there is no programming phase to transfer a bitstream from an external flash part.

[spilk]

some have the configuration memory on-die as well to keep part count down. Also good to help protect your bitstream, but this isn't infallible of course.