Comets have been observed for the last 2,000 years. During that time, several different systems have been used to assign names to each comet, and as a result many comets have more than one name.
The simplest system names comets after the year in which they were observed (e.g. the Great Comet of 1680). Later a convention arose of using the names of people associated with the discovery (e.g. Comet Hale–Bopp) or the first detailed study (e.g. Halley's Comet) of each comet. During the twentieth century, improvements in technology and dedicated searches led to a massive increase in the number of comet discoveries, which led to the creation of a numeric designation scheme. The original scheme assigned codes in the order that comets passed perihelion (e.g. Comet 1970 II). This scheme operated until 1994, when continued increases in the numbers of comets found each year resulted in the creation of a new scheme. This system, which is still in operation, assigns a code based on the type of orbit and the date of discovery (e.g. C/2012 S1).
Before any systematic naming convention was adopted, comets were named in a variety of ways. Prior to the early 20th century, most comets were simply referred to by the year when they appeared e.g. the "Comet of 1702".
Particularly bright comets which came to public attention (i.e. beyond the astronomy community) would be described as the great comet of that year, such as the "Great Comet of 1680" and "Great Comet of 1882". If more than one great comet appeared in a single year, the month would be used for e.g. the "Great January comet of 1910". Occasionally other additional adjectives might be used.
Possibly the earliest comet to be named after a person was Caesar's Comet in 44 BC, which was so named because it was observed shortly after the assassination of Julius Caesar and was interpreted as a sign of his deification. Later eponymous comets were named after the astronomer(s) who conducted detailed investigations on them, or later those who discovered the comet.