AKA Red Planaria. Others have described them pretty succinctly.
The best wrasse I've found for them is the Melanurus. Some yellow or green Coris will too, but I had no success with a six-lined wrasse. Melanurus is a flatworm eating MACHINE. As long as your tank is large enough to give one ample swimming space, that would be my recommendation.
On coral colonies that you are able to remove from the tank, a quick swirly in fresh RO for 5-10 seconds will kill them - they can't take the change in osmotic pressure and they die and drop off pretty much immediately. However, once they are in the tank it's hard to get rid of them without a natural control.
There are "chemical weapons" against them, about 10 years ago I used Flatworm eXit in my 120 that had a severe infestation, and I ended up losing 7 of 13 fish. It wasn't the eXit's fault, it was the sheer volume of dying flatworms and the toxins they were releasing as they died. I followed directions to the letter and had more supplementary filtration than recommended but my outcome was still disastrous (corals were fine but the fish ... I still mourn those losses). If you treat chemically, if at all possible, remove your fish to another tank beforehand, especially if you have a really severe infestation.
Jenn