PostScript is a language that tells the printer how to output the file you are sending to it. PostScript is the de facto standard in the printing industry and is preferable because of its high-quality output of even complicated documents.
If you have printed a document from any Adobe program, not just Illustrator, you have experienced PostScript printing without really thinking about it. This goes for printing to high-end devices at an output agency or printing to the laser printer in your office. For the most part, you shouldn't have to put a lot of thought into PostScript, unless you suddenly have problems printing your documents and generate the dreaded PostScript error.
If you do have problems with PostScript printing, the following are some things to try to get through the issue:
Know that when you are preparing documents to be output, your printer may supply you with a special PPD that prepares your document for output, especially if you are creating PDFs. To load a specialized PPD, select Other from the PPD pop-up menu in the Print menu and then browse to the provided PPD and click Open.