Soft coral is a general term used to describe the order Alcyonacea. The hobby tends to differentiate these corals by their lack of skeletons. While it's true that unlike LPS and SPS corals these animals do not secrete calcium carbonate over time to build a stony skeleton. They do, however, have spiny skeletal elements called sclerites, these are used to give the colony support and cause the flesh to have a grainy texture. Gorgonains also have flexible skeletons formed by a complex protein called gorgonin.
The term Leather Coral describes the family Alcyoniidae a subset of the order described above. Some common leather corals you'll encounter in the hobby are Toadstools, Devil's Hands, and Finger Leathers.
Leather corals will periodically shed a thin layer of their tissue to rid the coral’s surface of waste, debris, and algae. While this is happening the coral will look very unhappy for a few days. This can be alarming to a hobbyist who has yet to experience this, however, it's a completely natural process and is actually a good sign that the coral is healthy. It's not uncommon for shedding to occur when moved to a new system. This process can take up to a week in a lower flow system, the more flow provided the faster it will shed.
Often people hesitate when it comes to adding soft corals to their reefs as they've heard stories of them becoming invasive and taking over the tank. While this can certainly happen with some soft corals, it's the exception not the rule.
This is only really the case for Pulsing Xenia, Green Star Polyps and Kenya Trees (Capnella or Nephthea). These corals when given their ideal conditions will grow fast and can take up a large portion of your aquarium quickly. If you want to avoid this we recommend isolating them on their own rock island. This allows you to have a small Pulsing Xenia island without the risk of it slowly covering your entire rock structure.