I am sure others will have something to add, but here are the basics.
Soft corals are things like mushrooms, zoas, leathers, star polyps, etc. These corals do not have their own skeleton and basically attach to the rock or substrate with thier flesh. Some grow in the form of a mat (star polyps) while others are more solitary but may form new individuals next to existing ones (mushrooms). These corals require good water quality, but are the most forgiving of water quality issues normally. They also normally require moderate light levels (compact florescent or higher)
LPS corals are what are referred to as Large Polyp Stony corals. These corals have their own skeleton that as they grow they build by utilitizing calcium and carbonate (among other things) out of the water column. They can be branching or more solitary. They have large fleshy polyps that will inflate during the day and deflate some at night. Some LPS corals include brains, candy cane, torch, hammer, acans, plates. There are many more, but that gives you an idea. These corals require good water quality, but will tolerate some deviation from ideal in most cases. Most of these corals require moderate light levels although some require more light than others. A lot of these corals cannot tolerate a ton of flow because of the large fleshy polyps that they have.
SPS are Small Polyp Stony corals. These are the hardest corals to keep generally. These corals also build a stony skeleton, but unlike LPS the polyps are very small and normally you would not consider them fleshy. Examples of SPS corals would include Acropora, Montipora, Birdsnest, Millepora, Pocillopora, ect. These corals require excellent water quality and high light (Metal Halides or individually reflected T5s are necessary in most cases)and high flow.
