A descriptive word or phrase added to substituted for a persons name?

You're looking for a descriptor or epithet, which is a descriptive word or phrase used to replace a person's name.

Here are some examples:

Descriptors:

* The Professor

* The General

* The Queen

* The Detective

* The Doctor

* The Architect

Epithets:

* The Great Liberator (for George Washington)

* The Iron Lady (for Margaret Thatcher)

* The Bard (for William Shakespeare)

* The King of Pop (for Michael Jackson)

* The Queen of Soul (for Aretha Franklin)

Other possibilities:

* Nickname

* Title

* Occupation

* Physical characteristic

* Personality trait

The best descriptor or epithet will depend on the context and the person you are referring to.