The 8259A Programmable Interrupt Controller allows for 8 interrupt inputs to be handled individually. It can prioritize interrupts and mask lower priority interrupts while higher ones are serviced. The controller's pins include data bus pins to transfer control/status information, as well as pins for chip select, read/write, cascade connections for multiple controllers, and interrupt request/acknowledge lines. It provides flexible interrupt handling capabilities for microprocessors like the 8085A and 8086.