An FPC is a flexible printed circuit also known as a "flexible PCB" used to protect and manage a lithium-ion battery pack. It is a type of PCB (printed circuit board) or PCM (protection circuit module). An FPC is typically used for flexible devices, for example, wearables like smartwatches, or medical monitoring devices which use the lithium polymer, or LiPo battery.
These flexible PCBs are also used for NFC (near field communication) devices. In a smartphone, the NFC chip often accompanies the battery because it is an ideal location to communicate from, for example, through file transfer or payment transactions on a POS system.