In short, the standard name is the official name of the species. Common names are all the other names that have been used. In Australia there have been over 13000 common names used for around 4500 species!
Perhaps the best way to clarify this is with an example. The Snapper, Pagrus auratus, goes by many common names in Australia;
Cockney, Cockney Bream, Eastern Snapper, Nobblers, Old Man, Old Man Red Snapper, Old Man Snapper, Pink Snapper, Pinkie, Pinkies, Queen, Red Bream, Reddie, Ruggers, Schnapper, Silver Seabream, Squire, Tamure, Western Snapper, White Snapper
but only one standard name;
For more information on fish names, go to the Australian Faunal Directory. This excellent site lists the standard name (in red) followed by the common names, for each Australian species. In addition, information on the species' taxonomy and distribution are also provided. The page for the Snapper is here.