What does "DeWitt W. Buchanan, Jr." in a professor's title mean?

You would list the "DeWitt W. Buchanan, Jr., Distinguished Professor of Dispute Resolution and Organizations" if you were, for instance, issuing a press release about Prof. Brett's accomplishments, or writing very formal correspondence. You might also mention it if you were introducing Prof. Brett in a seminar.

Under most normal circumstances, however, such information is not needed, and could come across as being almost pretentious.

As for what the "Dewitt W. Buchanan, Jr., Distinguished Professor" represents, it's what's known as an endowed chair or named chair. Mr. Buchanan was the person honored by the chair—either because he gave the money to establish the chair himself, or others gave the money on his behalf or in his honor.

It looks like the professorship she holds was named in honor of Mr. Buchanan. You shouldn't need to include it when you cite her.

This is a so-called "chair" professorship, where the "chair" was endowed by a wealthy man named DeWitt W. Buchanan Jr.

That means that the donation given by Mr. Buchanan Jr. was large enough to pay the holder of the "chair" over her probable tenure, so he gets to have his name attached to the work of that one professor.