It's called a "byname" or "epithet". It's an old-fashioned version of last name before the system of hereditary fixed surnames came about. Sometimes, a person's children would use their father's byname if it applied to them as well, but that usually didn't last more than a generation.

A byname was a literal description of the person: "the Red", "the White", "the Black", and "the Brown" were usually an indication of ginger, blonde, or brunette hair color, though red and white could also refer to complexion. (Someone dark enough to be called "the Black" or "the Brown" in England based on their skin color and not hair color was usually called "the Negro", though as you might imagine, these instances were few and far between in old England.)

I have written about the history of European last names in this sub before here with some mentions of "the Red". You can find sources and further reading at that link.

Long story short, fixed hereditary family surnames emerged shortly after the Norman Invasion in both England and France, and then the English dropped the prepositions from last names during the Hundred Years War for unknown reasons, possibly to differentiate themselves from the French. So when the transition occurred from bynames/epithets to modern surnames, English people ended up with the surnames White, Black, Brown, and Rudd.

In other European languages/cultures, the timeline of the switch from bynames to fixed surnames (most with pit stops using a strict patronymic surname system in between) happened over a very long period of time, between 1100 when the English and French first began doing it, to 1900 when Sweden had finally made the change. (And Iceland still never has switched over, continuing with a patronymic surname system.)

So you also see these kinds of last names in other languages. For example, the versions of "the White" from other European languages include "Weiss" (German for "white"), "Dewitt" (Dutch for "the White"), and "Le Blanc" (French for "the White").

You may also ask, "What about the last name Green/Greene? That can't mean green hair, can it?" Bynames/epithets weren't always based on personal descriptions. In fact, it was more common for your byname to be based on your occupation or the location of your house. Names like "Wood", "Atwood" and "Hill" started out as "John at the Wood" and "William on the Hill". The last name "Green" is the same way. It was an indication of someone who lived near the village green, i.e. near the town square. "Henry at the Green" became Henry Green.