esentially think of it as there are four types of corals you can buy (please don't flame me, I've simplified this greatly):
1) zoanthids, leather corals, clove polyps, etc.
Known as sea mats, zoanthids are a colony of single gut organisms which spread over a previously deposited surface. They in no way deposit their own support strutures.
2) sea fans, gorgonians, etc.
For the most part they are passive filter feeders, some able to orientate themselves within the water column to maximize their ability to collect food from the water column. they have individualized septa which aids in the segmentation of their bodies. They also lay down and inner skeleton of CaCo3, and CaCO3 spicules to help support their bodies. But they Do Not lay down a fully CaCO3 skeleton like the next two groups.
3) Hammer corals, plate corals, torch corals, open brains, acan's etc.
LPS means " Large Polyped Stoney" coral. These corals actively deposit CaCO3 underneath their tissues to buils a sturdy base to grow and epand from. They can usually inflate or deflate their tissues to aid in light absorption, prey capture, self defense. How can you tell them apart from SPS? Usually, as the name implies, the most simplistic but not always accurate way is to look at the size of the individual polyp. If you can easily make out a polyp without squinting, chances are it's an LPS. Also most SPS will not inflate in the same manner LPS do. The SPS will appear, as stated above, look like colored sticks.
4) Acropora, montipora, etc.
SPS or "Small Polyp Stoney" is as the name implies, small polyp coral. Usually SPS corals look rigid. The tissue is less expanded than with LPS.
Like I said......over simplified....not very correct, but may help a little.