The term black belt is almost universally understood. Everyone knows someone who is black belt.
The term black belt is often misunderstood to mean absolute expert. A black belt, according to this misunderstanding, is supposed to be able to defend against 12 people, levitate in the air, have magic superpowers, and be all seeing, knowing, and understanding. This is not realistic. There are often a variety of myths associated with being a black belt, such as you hands and feet are deadly weapons, and the black belt has to register themselves as a black belt with the government.
A black belt is a person who knows the basics of their martial art, and is ready to start a serious level of study. They reasonably should be able to perform their art against a person of equal weight and size. A better way to put it, the black belt should be much more practiced and ready to perform their martial art, better then they were before they started.
A legitimate black belt should also have a mature, honorable quality to themselves, produced through the perseverance required to take themselves to the black belt level. As is often said about black belts:
* A black belt is a white belt that never quit.
* A black belt is a leader.
* A black belt doesn't know the word can't.
The fact of modern martial arts practice is that the majority of students are children. This leads to children taking lessons long enough to earn themsleves a black belt, referred to as junior black belts. There is a fair amount of concern about this situation from traditionalist thinkers, as an eight year old can hardly cross the street, much less have the maturity to deal with marital/potentially dangerous fighting methods. The junior black belt is meant to convey a less mature, not yet adult designation.
The black belt has a lot of different meanings, as different arts and individual schools use it for differing reasons. In some schools the black belt is difficult to obtain, takes a long time, and is really an indicator of achievement. In other schools everyone who takes lessons for 2 1/2 - 3 years receives this (and these type of schools graduate a lot of people). And is other arts, the black belt (sash) is the highest level one can obtain, or, as with Brazilian Jiu Jitsu, it takes 10 years to receive this honor, and means expertise.
No other concept in more important in modern martial arts as the black belt. This is the one thing that every starting students wishes to achieve, even though the great majority do not persevere.