A generator rated for "Prime Power" (ie. the main source of power for a given load) is a 24/7 continuous use duty rating.
"Stand-by Power" is just that, and is often a 12, 24, or other hour "surge rating"
Stand-by/Emergency power are often the same things, however there are some varying definitions depending on the application. (Ie. a standby generator may actually come on when grid power is still on to meet excess demand, and an emergency generator may be an additional generator that controls only "life-critical" loads)
You will often see the same generator rated with Prime/Standby ratings. The Prime rating is often about 10-25% lower than the standby rating.
See examples of differnt ratings on this spec sheet:
http://www.generac.com/PublicPDFs/0169500SBY.pdfSteve