"Black Death" refers to an epidemic of disease in the medieval world. Sometimes it just means the specific outbreak that spread west from Asia and peaked in Europe in 1346-1351. Sometimes it also includes subsequent waves of epidemic throughout the rest of the Middle Ages and into the early modern era.

Medieval sources frequently call the disease "pestilence." Looking at modern outbreaks of bubonic plague (including San Francisco in 1900 and 1904!), historians used to assume that pestilence was bubonic plague spread by "Fleas on Rats". Bubonic plague is one form of the illness caused by the bacterium Yersinia pestis.

In the 1990s in particular, several prominent historians went back to medieval descriptions of pestilence and made several key observations. Most importantly, the descriptions are sometimes really, really different. It's rd to arrive at a united view of what "pestilence" was based on chronicle accounts alone. Anthrax was proposed as a possible alternative, among others.

In 2010, a study by Haensch et al. published in PLoS Pathogen found concrete DNA evidence that "pestilence" was, indeed, caused by Y. pestis (or rather, medieval variants of it). Plague comes in multiple forms, although bubonic is the most famous and distinctive. Something like pneumonic plague that attacks the lungs and kills very quickly is also attested in medieval sources.

Historical epidemiology is hard, because diseases do evolve over time. Symptoms can change as well as people's general susceptibility. But the "state of the question" in medieval history right now is that Y. pestis is pestilence.

If you're interested, Rosemary Horrox has published a really neat anthology of primary sources related to the Black Death translated into English, The Black Death. It includes chronicle and literary descriptions of the plague as well as proto-scientific writing on how to combat it. It's fascinating reading.