An epithet is a word or phrase which is used to characterize a person or thing. For example, the term "President" in “President Washington” is an epithet, because it provides specific information which makes the identity of the Washington under discussion obvious. The usage of "epithet" to describe a derogatory slang term is common in some parts of the world, and it is considered technically correct by most people.
In biology, the term "epithet" has a very specific meaning, although this meaning is related to that which is in more general usage. A biological epithet consists of the species of an organism when it follows the genus, precisely codifying it so that everyone knows which organism is being discussed. For example, in Sequoia sempervirens, "sempervirens" is the specific epithet for the coast redwood tree. Specific epithets help to clarify biological nomenclature, ensuring uniformity in the field and reducing the risk of confusion.
You can find many examples of epithets in history books. For example, famous leaders are often identified with an epithet, as is the case with people like Alexander the Great, Richard the Lionhearted, and Joan of Arc. Epithets are also common to many religions; the Dalai Lama, for example, is formally known as His Holiness the Dalai Lama, while religious officiants are identified with terms like “reverend” and “father.” Epithets were also historically embedded in many names, and you can find examples of such epithets in last names like Cooper, Smith, and Gardener.
Being aware of the proper use of epithets is often an important part of etiquette education, as some people grow quite upset when you refer to them incorrectly. For example, many people with a PhD degree like to be called “Doctor,” and it helps to know that dukes are addressed as “Your Grace,” while royalty generally prefer to be called “Your Highness” or “Your Royal Highness.” In some cultures, failure to use an appropriate epithet is considered extremely rude, even if the error is made out of ignorance.
Using the term “epithet” to describe many insults actually makes sense, because insults are often based in descriptions. Racial insults, for example, often play on slang terms used to describe people of specific religions of races, and adjectives like stupid are often used with the intention of insult.