The disease - later called the Black Death (because of the dark patches on the skin caused by subcutaneous bleeding) was probably Bubonic Plague. Bubonic plague is carried by black rats, and spread to humans by the fleas that infest them. Human carriers could themselves pass on the infection through contact or the air.
The rapidity of the disease's spread (not characteristic of modern outbreaks) and the lack of contemporary reports of numerous dead black rats suggests that other diseases (for example typhus) may also have been involved.