EDIT: here is a "better" more basic answer.
Soft corals or "softies" are corals that do not make a calcium carbonate skeleton (are NOT "stony"). Some of these corals produce a substance called gorgonin in their bodies and they are called gorgoania/gorgonians. Soft corals are also octocorallia (have 8 tentacles per polyp/mouth).
Stony corals are corals that produce a calcium carbonate skeleton, they are sometimes called "true corals" or "reef building corals". Stony corals are also hexacorallia (have 6 tentacles per polyp/mouth). Hobbyists and the aquarium industry break stony corals down into 2 basic groups based on their relative and subjective polyp size. Those corals with large polyps are called large polyp stony or LPS and those corals with small polyps are called small polyp stony or SPS. Both LPS and SPS are further broken down by a number of factors including polyp/tentacle shape and growth pattern.
Note that in this hobby it is not uncommon for trade names, common/hobbyist names, and scientific names (both correct, incorrect, outdated and up to date) to be used in conjunction with each other and interchangeably. Example might be: Hobbyist says "that is a nice flowerpot coral, is that a goni or an alveopora?" and the store owner replies. "that a 'Miss Piggy' goniopora, it is one of my favorite LPS"