According to Wikipedia, it's known in modern times as mycobacterial cervical lymphadenitis.

The disease mycobacterial cervical lymphadenitis, also known historically as scrofula and the king's evil, involves a lymphadenitis of the cervical (neck) lymph nodes associated with tuberculosis as well as nontuberculous (atypical) mycobacteria such as Mycobacterium marinum.

As for what lymphadenitis is?

Lymphadenopathy or adenopathy is a disease of the lymph nodes, in which they are abnormal in size or consistency. Lymphadenopathy of an inflammatory type (the most common type) is lymphadenitis.

The Wikipedia article for king's disease has a picture if you're curious, although it's a bit gross.