<< what would some real world applications be for the ASIC? >>
Both ASICs and FPGAs can be used for any of the following (and a lot more): digital camera chips, network router, engine/brake control system, hard drive controller/interface, televisions, PDAs, printers, custom 3D graphics generation, name just about anything electronic and they're in there.
Also, the phrase ASIC/FPGA hardware engineer can be a little deceiving. Igor's definition is the best, but that doesn't mean that companies use that right. Sometimes they describe people who do strictly the chip, and sometimes the chip and the rest of the circuitry and board that the chip goes on.