The 8259 Programmable Interrupt Controller (PIC) is used to expand the number of interrupts for processors like the 8085. It can manage up to 8 interrupt requests at once and allow them to be serviced one by one. The 8259 can be programmed to determine interrupt priorities and masking. It is interfaced with the 8085 through I/O mapping and provides the interrupt acknowledge response by outputting the call opcode and address.