What's the term given to names with "The" in the middle of them, for example; "Robert The Red" or "James The Great"?
From the definition of epithet (from the Greek epitithenai, to add on) in Merriam-Webster's Vocabulary Builder By Mary W. Cornog:
A descriptive word or phrase occurring with or in place of the name of a person or thing.... Sometimes an epithet follows a given name, as in Erik the Red and Billy the Kid.
The definite article specifies the particular person under consideration. There may be many Eriks, but only one is ruddy enough to have Red attached to his name as an epithet.