Epithet

An epithet is generally used in order to attribute a certain trait to an entire group or category of people. In modern usage, the most common type of epithet has become, unfortunately, the racial slur. This is likely based on the fact that when one does attribute a characteristic to an entire group, one usually relies on prejudices in order to do so. An epithet could technically attribute something positive to people as well, although this is far less common.  

Negative connotation

The term epithet literally means "attributed"; this is reflective of the fact that the term refers to an adjective that defines a person or thing. The term epithet has somewhat negative connotations, referring to prejudice and the causes of racism. But this need not be the case. An epithet can also define a person or thing in terms of a positive attribute.  

Using epithet correctly

In the event that you are still a little confused about the meaning of the term epithet, here are some examples of the term being correctly used. 

"In light of a more enlightened moral analysis of the actual actions of Alexander the Great, some activists began to call into question whether it really was appropriate to use the epithet "great" to characterized a person who was responsible for so much violence in the world."

"The man resented the racial epithet that was used to describe him, on the grounds that he found it dehumanizing: it negatively identified him with a group of people, without any real consideration for his own personal qualities as a human being." 

"The poet pointed out that a standard and socially accepted epithet for describe an object, such as calling a casino 'sleazy', is often based on prejudice and lack of imagination, insofar as the epithet tends to cause people to rely on stereotypes as opposed to trusting their own concrete experiences.  

For your reference, here are a couple rules you can follow in order to make sure you are using the term epithet correctly . 

1. An epithet is always an adjective which characterizes a person or thing and seeks to distill his entire being into that one attribute. In a way, then, epithet is related to metaphor and synecdoche. For example, in the name Alexander the Great, the clear implication is that greatness is fundamental to the very essence of what Alexander is. 
2. Due to its very nature, however, the epithet can often have a negative connotation. For example, in its most common contemporary usage, it has come to refer to racial slurs, or identifying a given person with a derogatory term used to refer to his entire "class" of people. Likewise, a person may choose to reject an epithet attributed to him, on the grounds that it is either reductionistic or insensitive.  

Used in psychology

Psychologically, the epithet is clearly related to the impulse to distinguish the most salient feature of a person or thing on the one hand, and then to honor that feature on the other. For example, when the epithet of "the Great" is attributed to Alexander, this indicates on the one hand that Alexander's fundamental nature has been identified, and on the other that the man is not just one Alexander among many others. 

It would be interesting in this context, perhaps, to trace the historical factors that has led to what could be called the degradation of the epithet. However the epithet first began, it is has increasingly been used to identify not an exemplary good feature in a person but rather an exemplary bad feature. This has become the case to the point that it has even become unusual for a given writer to use the term epithet in a positive way in contemporary times.