The answer provided already is satisfying but gives only one unique property of carbon.
One, as already mentioned, its ability to form various types of bonds as it is found in inorganic materials, but also is the very base of organic sciences.
The second I would say, is that it forms the 2 most opposite materials found, diamond which is unbreakable under normal conditions and also graphite which is super weak and fragile. and the only difference to make them is the condition of temp. and pressure that they were created in. Knowing that graphite is one of the best materials used in nano-sciences for its great shear-tensile behavior.
The third one will be its abundance I believe.
I also believe that someone else might say different 3 points depending on how they use or study Carbon or their background of its wide uses.
