If a cancerous cell enters the body of a healthy person from someone else's blood or something, will that healthy person get cancer? In human beings.
Can a cancer cells from someone else's body cause cancer in a healthy person?
No. Cancer cells from another person cannot cause cancer in a healthy person. The rare cases of transmissible tumors all involve unhealthy or not yet developed persons.
Transmission of tumor cells from one individual to another happens, but is quite rare, and in all cases involves some compromise or reduced development of the immune system. Though tumor cells do metastasize in an individual, when this occurs, tumor seeds must be able to evade the immune system and find an environment suitable for adhesion and replication. Tumor associated cells (non cancerous cells that regulate the microenvironment to make it favorable for growth and replication) are discussed in this seminal paper on cancer biology by Hannahan and Weinberg. There are similarities to infectious processes, but cancer is not measles. Tumor cells don't shed in comparable numbers, aren't adapted for immune escape in a separate host, and don't express appropriate adhesion proteins for portals of entry on a new host or readily induce tumor associated niches in a new host. The cases where person-to-person transmission of cancer via tumor cell inoculation does occur seem to demonstrate more how cancer cells are not infectious agents.
Donor-related tumors in transplant patients occur in immunosuppressed patients, but are still rare. The low frequency of transmission seems to be due, in part, to screening. The fact that we see this at all demonstrates the significance of transmission route and immune escape.
Maternal-fetal, and in utero twin-twin seem to be exceedingly rare, but have occurred, again, demonstrating the existence, but poor efficiency of transmission. Here, the fetus has an undeveloped immune system. I would not consider this case to be cancer cells causing cancer in a healthy person.
Inoculation of volunteers with tumor cells in a problematic series of experiments at Sloan Kettering in the 50s, transplantation of tumor cells into patients with other cancers, resulted in growth, recurrence after excision, and death in some cases. Transplantation into healthy volunteers (yes, they did this) resulted in nodules that spontaneously regressed. This experiment has since been interpreted as evidence for immune system control of transplanted tumor system in healthy individuals, as compared to growth and progression in a receptive niche in a cancer patient.
So person-to-person transmission of cancer cells is rare and requires an immunosuppressed or undeveloped host, or a host who already has cancer. There are no documented cases of person-to-person transmission to a healthy individual, and documented cases of failed transmission despite a surgical attempt. This is because, unlike an infectious microbe, in a healthy individual, there is not a suitable receptor for adhesion at an exposed or accessible site, a suitable environment for replication, and adaptations for immune escape by tumor cells in the original host are not effective in a new host.
As a side note, there are contagious cancers in other species, but this doesn't seem to be particularly relevant to a question about whether cancer can be transmitted between two humans. Many cancers have transmissible risk factors (e.g., human herpesvirus-8, hepatitis B and C viruses, human papilloma virus 16 and 18, and others)
Before OP edited his/her question, it was a little unclear whether the question was only about humans. The following answer is more general than asked as it also considers cancers in non-humans
Most cancers are not transmissible but some are. We call them (clonally) transmissible cancers.
The most famous case of transmissible cancer is the Devil facial tumour disease in Tazmanian devils. Other cases of transmissible cancers exist in Syrian hamsters, dogs (CTVT), and some bivalves. No such transmissible cancer is known to exist in humans.
Transmission of viruses inducing cancers
There are cases of cancer caused by viruses. Those viruses are transmissible and hence it looks like the cancer itself is transmissible. In humans, this is for example the case of Kaposi's sarcoma-associated herpesvirus.
Transmission via transplant
Finally, there are cases of cancer that can be transmitted to a new person via a tissue transplant. In humans, Kaposi's sarcoma is (again) an example.
The answer is no.
From molecular/mechanistic point of view cancer cell itself is not a self-maintaining and independent creature. Even assuming its infinite potential to renew and grow, it still needs constant nutrients supply and tailored microenvironment to survive. Therefore, there is a growing number of attempts to therapeutically target tumour cells niche, see: Joyce JA, Cancer Cell, 7(6), P513-520, 2005 and Belli C. et al., Cancer Treat Rev. 2018 Apr;65:22-32. To overcome this limitations, Cells can undergo epithelial-mesenchymal transition, which allows for migration outside the primary location and is crucial to initiate metastases.
Another important issue is the major histocompatibility complex, which allows to identify and eliminate cells, that contains foreign antigenes. Also, constant immune surveillance in immunocompetent individuals leads to elimination of potential cancerogenous cells. Therefore, patients with immunodeficiencies (eg. with AIDS or on prolonged pharmacological immunosuppression) have significantly higher risk of developing infection-related cancers, such as cervical cancer (HPV) or Kaposi Sarcoma (HHV-8).