Others have already given many great examples, but here's a sample. The mechanisms by which a cancer can cause death can be classified like this (just a convenience classification, others are possible):
1) By becoming large enough to cause pressure-related complications locally. This applies especially to confined, anatomically critical locations like within the skull, so aggressive brain tumours (like glioblastoma multiforme) kill this way. Sometimes, even when there's space to grow, the tumour may obstruct the function of surrounding healthy cells and ducts, and cause a slower death that way, e.g. pancreatic cancer causing biliary system obstruction and all its attendant complications.
2) By causing systemic (body-wide) complications related to hormone-like effects. These are called "paraneoplastic syndromes". A serious one is related to a marked rise in serum calcium, that can have effects on the heart. Another serious one relates to an increase in clotting tendency, that can cause clots to form in the vascular system. The clots can travel to the vessels of the lung, a complication called pulmonary embolism, which can lead to compromised respiratory function and sudden death.
3) By compromising the immune system. This can happen directly, like in blood-cell cancers where the organs that generate immune system cells are directly shut down and the progenitor immune system cells are crowded out. It can also happen indirectly, like when chemotherapy given to cancer patients also hit healthy bystander immune cells, causing complications like neutropenia (markedly reduced numbers of a particular fraction of white blood cells) that can lead to a serious susceptibility to life-threatening infections.
Those are the main modes by which death occurs in cancer sufferers. Of course, rarer complications can always occur, for example in a large and friable (fragile) tumour, a piece can easily break off and block a large vessel or break into smaller pieces and embolise to the lung vessels. And sometimes cancers be associated with certain infections that can then cause further infectious complications. An interesting example of the latter is with a group of bacteria that are loosely known as Streptococcus bovis group, which have a tendency to cause colorectal (large intestine) cancers. The cancers themselves can also increase (by destroying the normally ordered tissue architecture) the likelihood that these bacteria migrate into the bloodstream (called translocation) and infect the heart's valves, resulting a severe infection called endocarditis, which can be life-threatening. But by-and-large, these are just interesting oddities that are nice to know about (unless you work in the field, as I do).