These are the easiest corals to keep in your reef aquarium and ideal for the beginner. These corals generally have a leathery texture to them and have no skeleton structure to them. They rely on photosynthesis to grow. These and many others corals have an algae present called zooxanthellae. Both the coral and the algae rely on each other, if one was not present the other would simply die. So in order to keep these corals you will need lighting such as T8's or T5's will be more than adequate.
Examples of Soft corals include: zoas, button polyps, mushrooms, ricordias, toadstools and more.
These make up the majority of the remaining corals but come in two forms LPS and SPS. Large polyp stony and small polyp stony. Both types consist of a skeleton type structure this makes them much more demanding and require stable water parameters in magnesium, alkalinity and calcium. without this the coral will simply die and all that will remain is the skeleton.
LPS or Large poly stony corals are some of the nicest corals available with amazing colors and interesting features. These corals Have a fleshy appearance some have tentacles, some almost flower like in appearance, many have branching formations. Some just appear flat and have little if no movement at all.
Most LPS corals require feeding as well as stable conditions. Good lighting is also required. T5 minimum right through to metal halide, although many reefers are now keeping these corals under LED lighting. Medium to strong flow is best for these corals.
Examples: Torch, hammer coral, Duncan's, brains, scollys, acans, favia, and many more.
SPS, Small polyp stony corals have a skeleton like LPS, but have tiny polyps on their tree, branch like appearance. These corals require strong lighting like LPS but need strong flow to help them thrive. You will soon learn when keeping these that extra equipment such as a calcium reactor or balling light salts will be needed to keep up with their consumption of alkalinity and calcium. If a newcomer to the hobby i would avoid these until you understand these corals and the methods for maintaining their needs.
Examples: Acroporia, montiporia, and more.
To conclude this article, I would have to suggest starting at the top and working you way down as experience and understanding progress. This is a simple, tip of the iceberg type article that should help guide your way to the basic types of corals available. There are many more that maybe covered in future articles.