Chow ring
One application of this is to define intersection products in the Chow ring. Suppose that "X" and "V" are closed subschemes of "Y" with intersection "W", and we wish to define the intersection product of "X" and "V" in the Chow ring of "Y". Deformation to the normal cone in this case means that we replace the embeddings of "X" and "W" in "Y" and "V" by their normal cones "C""Y"("X") and "C""W"("V"), so that we want to find the product of "X" and "C""W""V" in "C""X""Y".