LPS - Large
Polyp Stony coral. These corals have a hard calcium skeleton with large fleshy polyps. Reguire medium to strong light, high/stable calcium levels and gentle to moderate current. eg Hammer, Frogspawn, Fox, Candy Cane, Maze, Bubble corals to name a few.
SPS - Small Polyp Stony corals. Cacium skeleton/small polyps. Most require strong light, high stable calcium levels, and stron random currents. Water conditions must be and remain top notch. eg Acropora, Montipora, Pocilliapora, Pavona, Turbina, etc.
Soft Coals are non-reef building corals, lacking a calcium skeleton. Most are hardy and can survive in less than perfect conditions and are therefor great begginer corals. Low to medium light and current. eg Leather, Xenia,
Green Star Polyps, Colt, etc.
This is of course a very general description, each coral will have its own prime conditions and those may vary from group to group. Not all SPS corals are hard to keep and not all soft corals are easy to keep. Its best to do your research before you buy or attempt keeping these animals. A good book will help. Corals: A Quick Reference Guide by Julian Sprung and Aquarium Corals by Eric Borneman are great books to get you started.
HTH