Angiogenesis is the growth of new blood vessels from preexisting blood vessels. Whilst this process is normally positive (e.g in embryo maturation) it is a key requirement in the etiology of cancer and tumour formation and growth.
Angiogenesis in this case is the proliferation of a network of blood vessels that penetrates into cancerous areas. In its so called sprouting form the new vessels emerge stimulated by angiogenic growth factors, via endothelial cell proliferation. Whereas in other forms the core vessel split to form a new network.