Put simply, the epidemic known as The Black Death, Black Plague, Great Pestilence etc., was yersinia pestis passing through Europe from 1346-53. Yersinia pestis is a bacteria which can infect a person in three ways: bubonic, pneumonic, and septicaemic. I am not a biologist, so I can't explain exactly how yersinia pestis kills a person and can only explain it in broad strokes.

Bubonic Plague spread through the bite of fleas. Through the skin, yersinia pestis binds itself to white blood cells until it infects the lymph nodes, which are little organs which produce white blood cells and antibodies. Once there, the bacteria is surrounded by the chemicals it needs to survive and reproduce quickly. As a result, the lymph nodes swell up which is where the distinctive buboes come from. The infection will then spread throughout the lymphatic system, damaging blood vessels which may cause gangrene as well as the coughing, excreting, and vomiting of blood. yersinia pestis binds to chemicals used by the lymphatic system to communicate, so not only is the body deprived of the organs used to produce white blood cells, but it doesn't know where to send the white blood cells it does have. Once the body realises that the infection cannot be stopped in this way, it will progress to sepsis. If the antibodies of white blood cells are a precision missile, then sepsis is carpet bombing, and the last resort of the immune system. It will throw everything at the disease, and damage organs and blood vessels in the process. This is especially bad with an infection of yersinia pestis because the blood vessels have already been damaged and the immune system is effectively blind, so it doesn't know where to direct all its bacteria-destroying cells. The victim suffers catastrophic organ failure and dies.

Septicaemic Plague is when yersinia pestis manages to reproduce in blood vessels without having to settle in a lymph node. It causes serious damage to blood vessels and leaches iron from red blood cells, resulting in cell death, serious blood loss from infected areas, and eventual organ failure. Septicaemic Plague is almost always a result of first contracting another form of plague, such as if Bubonic Plague got stuck in a blood vessel and began reproducing there. Septicaemic Plague acts as a pathway to the two other plagues, as bacteria in the blood can become dislodged and settle in the lungs and/or lymph nodes. On it's own though, it will kill through loss of bodily fluids, either directly or by triggering sepsis. It has a 100% mortality rate.

Pneumonic Plague is yersinia pestis in the lungs. This is the most dangerous form of plague because it can be spread by droplets (mainly sneezing) and doesn't need fleas to infect others. In the lungs, yersinia pestis directly attacks the respiratory system. The victim will suffer breathing problems and suffocate, though the infection may trigger sepsis before then. This form has a ~95% mortality rate and can kill in 36 hours. Even today, if it isn't treated within 24 hours of symptoms showing, the victim will die.

With the depressing medical stuff out of the way, we can discuss what made the Black Death so dangerous in 1348-9. Yersinia pestis had infected Europe and the Mediterranean before during the Plague of Justinian, but it had mutated since then. Through the great mercantile cities of Italy, the plague spread easily in bubonic form on ships (pneumonic would kill most of a crew before a ship reached the destination) and then spread inland in the spring. Some cities attempted to quarantine themselves, but it usually didn't work. Once in the cities it could become pneumonic and spread rapidly throughout the population. As Flemmish monk Lodewijk Heyligen noted:

It is said that the plague takes three forms.* In the first people suffer an infection of the lungs, which leads to breathing difficulties. Whoever has this corruption or contamination to any extent cannot escape but will die within two days.

This is reinforced by the exhumation of 4 bodies in London in 2014. Bubonic Plague was definitely present, but every body tested had plague in the pneumonic form. The exhumation of 25 bodies in 2013 (also in London) also showed that the bacteria had mutated again into a form which is still with us, and hit Madagascar in 2013. This means that there were two forms of yersinia pestis going around Europe, each of which could cause 3 fatal diseases. These factors explain why the Black Death spread so well when it hit London in the late autumn/early winter of 1348. In winter, plague carrying animals are less active, and people stay indoors for longer, which should have slowed the rate of Bubonic Plague. London was also a rather clean city with less of a wild animal problem than places like Venice and Paris, with regular rubbish collection and street cleaning supervised by the city council. Bubonic Plague alone has never sufficiently explained the ~60% mortality rate that is thought to have occurred, because in winter in London it just shouldn't have spread so virulently (hence why we keep digging up plague victims to study them). But the city's relative cleanliness had little impact on Pneumonic Plague, and with the tests in 2014 we got confirmation that it was Pneumonic Plague which did the killing and the mystery was solved.

The Black Death killed so many because yersinia pestis was three fatal diseases rolled into one, and had mutated since the 6th century to be more infectious than ever, then mutated again as the plague went on so that even the people who recovered the first time weren't safe. It could spread in bubonic form on ships and through wild animals, travellers, and medics (often monks) until it reached a town or city and infected somebody's lungs. From there, it could wipe out half the population in a season. Then, the season after, the mutated yersinia pestis might arrive, as happened to London in the spring of 1349, and kill off even more people. We are not sure how many people the Black Death killed, and will never know. Before the discovery of Pneumonic Plague in London, it was generally thought that about 20-30% of England died, but recently estimates have crept up to 50-60% dead in about 18 months. Generally speaking, it is thought that about half of western Europe died in the plague.

*The other two forms were both bubonic, Lodewijk thought buboes of the armpits and the groin were two different illnesses, when in fact they are just the main locations where lymph nodes are clustered.