I have a black furry like growth on the branches of my plum tree.

That is black tree fungus.Use a systemic fungicide.

It is called Black Knot disease. It will eventually kill the tree. There is no cure and it could infect any other trees in your yard. The only thing to do is to cut it down. We had this same problem 2 years ago and after amputating the diseased limbs one by one we had to cut the whole tree down. Broke my heart. It was going to have a lot of Prune Plums a.k.a. Italian Plum tree. We recently planted a sweet cherry tree in its place.

Before you treat it check with an arborist to see if treating it will work or not and what they suggest.