I think for many people the experience of consciousness leads to an unshakable sense that there is something that exists which can be called consciousness and which is different from matter. Many philosophers deny this. For me the experience of existence leads to an unshakable sense that at least something exists but I can't say exactly what exists. I could say that my experiences must be caused by something but its conceivable that someone would deny that intuition. It wouldn't surprise me if some philosophers completely denied that anything at all exists. What is the name of that way of thinking? What philosophers have advocated that belief?

I am not aware of any philosophers that claim that absolutely nothing exists. The most skeptical position I am aware of is called solipsism, which is the view that only the self exists or that one is only justified in believing the self exists. However, it is a rather rare position. Descartes famous argument: "I think, therefore I am" has been pretty influential in establishing the existence of the self. However, even this rather intuitive argument has been appropriately criticized by Bertrand Russell insisted that all the argument really proved was that 'thoughts are being thunk.