I read another question where the author is asking about the infectivity of viruses. In the same vein, I am curious why cancer is not a communicable disease.
A cancer is not a pathogen
Cancer is a group of cells that (because of several specific mutations) start to duplicate abnormally. This group of cancerous cells are the own cells of the sick patient. It is not another species infecting the individual carrying the disease. It is the individual itself who developed the disease. Cancer is not a pathogen (but see below) that infect hosts and get transmitted from one host to another. It is more correct to picture cancer as an internal mistake of the body rather than as an external attack on our body. You might want to have a look to the wiki page for cancer for more information.
Transmissible agents that can cause cancer
Some cancers are caused by transmissible agents, such as oncoviruses. the transmissible agent may jump from one host to another and cause cancers giving the sensation that cancer is transmitted. It is just a sensation of course because in reality only the oncovirus gets transmitted and not cancer itself jumping from one host to another.
Cancers that become transmissible
Note however that it is not impossible to imagine a tumour jumping from one individual to another in cases where the immune response cannot detect cancer as 'non-self'. Some cancers are transmissible. Those cancers are called clonally transmissible cancer (As indicated by C Rags in the comments). Here is the list:
Naïvely, there’s no reason why cancers wouldn’t be communicable. Tumours consist of abnormally multiplying cells which have switched off their response to repressive stimuli from other cells, which would otherwise cause them to stop growing (plus a few other properties which have first been collectively described in the seminal paper The hallmarks of cancer).
And tumour cells can absolutely split off and start growing elsewhere, thus effectively spreading the cancer. This happens in metastases, and it happens in the few types of communicable cancer that are known.
So why does this normally not happen between individuals?
The reason is our immune system. Our bodies are normally incredibly good at recognising and eliminating foreign bodies – so-called antigens. The problem with cancers is that they are not foreign bodies: they are our own cells, only mutated. This makes it hard for our own immune system to recognise them as foreign. (It’s worth noting that at some point tumour cells have accumulated enough mutations to be recognised as foreign, but by this point the tumour is already mounting an effective resistance to the immune system.)
It’s for this reason that most cancers are not transmissible: invading cancer cells are immediately recognised as foreign bodies, and are efficiently eliminated by the immune system before they can multiply into a tumour. Unlike other antigens they have not evolved effective ways of evading the host’s immune system.
— Let’s take a detour here and talk about organ transplantation: When a donor organ is transplanted into a new host, the host recognises this organ as foreign, and initiates an immune reaction against it (or vice versa). This transplant rejection is a huge problem, and needs to be addressed by finding a closely related donor, and administering immunosuppressant drugs.
Which brings us back to transmissible cancers: all known types of transmissible cancer occur in species which have gone through a recent population bottleneck. All now-living individuals are very closely related. As a consequence, their immune systems don’t necessarily recognise another individual’s cells as foreign, or at least produces a reduced immune response.