There have been several pandemics of the disease called "the plague" (commonly identified, since the late 19th century, with bubonic plague, a disease caused by the agent Yersinia pestis) in history.
The first pandemic, the Justinian plague, began in 541 and devastated the Byzantine empire (and, presumably, much of Europe). It persisted until the 8th century, then died out.
The second pandemic, now known as the Black Death, began in Central Asia in the early 1340s, and reached Sicily in 1347. It continued to wreak havoc in Europe for centuries, only disappearing in the early 18th century.
The third pandemic likewise began in Central Asia, in the 1890s, and reached Hong Kong in 1894, where the Swiss microbiologist Alexandre Yersin isolated the agent of disease, a Gram-negative coccobacillus of the family Enterobacteriaceae now known as Yersinia pestis. The third pandemic is still active today, with periodic outbreaks in many locations around the world.
Of the three pandemics, the second was clearly the most disastrous. The first outbreak in Europe, from 1347 to 1351, killed approximately one third of Europe's population.
The term "Black Death", incidentally, was not used at the time of the second pandemic - it is a later phrase, a mistranslation of the Latin term atra mors (literally "black death", but correctly translated: "terrible death"). The mistranslation was probably made by a Danish or Swedish writer in the 1700s. The people suffering the effects of the second pandemic never used the term "Black Death" - rather, they called it merely "the mortality" (Latin: mortalitatis).
A useful page reference: www.scholiast.org/history/hi-bdth.html
An addendum: All of the above is a somewhat simplistic introductory text, which assumes that the identification made by Yersin and by the Plague Commission in India, identifying modern bubonic plague (the disease unquestionably responible for the third pandemic) with the Black Death (and, by extension, with the Justinian plague), is correct.
Needless to say, this is the subject of intense academic discussion - with historians, microbiologists, and medical doctors sometimes wildly at odds. Alternative hypotheses presented to explain the Black Death are that it might have been some other disease - smallpox, anthrax or typhus have been mentioned, none of them with much convincing evidence to back them up.
Recently, the researchers Susan Scott and Christopher Duncan (in the book The Biology of Plagues) have promoted an unidentified viral cause of the earlier pandemics of "plague", some sort of haemorrhagic virus (like, but not identical to, Ebola). They base this supposition on epidemiological/statistical analysis of verified outbreak patterns, which seem to suggest a much longer incubation period than is the norm for bubonic plague. Their book has generated even more heated debate, and continues to do so.
In sum, there are a lot of factual problems with the identification of the Black Death with bubonic plague - none of which are likely to be definitely resolved in the immediate future.