Well to put it simply, carbon forms the building blocks of life. Just about every living organisms is made up of some particular amount of carbon and every organic compound on the planet is made up of carbon. Carbon itself isn't especially useful, what makes it so important is the fact that it can merge with different elements and compounds in a multitude of different ways thus making it possible to form complex organic life forms.
If you removed carbon then it would mean that we'd be without things like fossil fuel(most used source of fuel on the planet), CO2(without it all plant life would die), and so many other things. Carbon is essential to life on earth, without it, there can be no life.