I don't understand what the term "double consciousness" means. I'm trying to figure how double consciousness applies to 20th century black culture...

Du Bois writes that African-Americans ("Negro" is the term he uses) are "born with a veil, and gifted with second-sight in this American world,—a world which yields him no true self-consciousness, but only lets him see himself through the revelation of the other world." In other words, one of the problems for the Negro/African-American is that he sees himself via the perception of white American culture. 

He goes on to write, "One ever feels his twoness,—an American, a Negro; two souls, two thoughts, two unreconciled strivings; two warring ideals in one dark body, whose dogged strength alone keeps it from being torn asunder." The Negro yearns to have a significant sense of his own Black and/or African heritage. But he is also an American, another significant part of who he is. The problem, in Du Bois' time, is that American culture was defined almost solely in terms of whiteness. So, the Negro is faced with this "double consciousness." It is a sense of belonging and not belonging. He notes how "American" culture is mostly a white notion and therefore a white identity. And he is forced to acknowledge his blackness via this veil of white American culture. 

The ideal solution is for the Negro to be both Black and American, without that Black identity being obscured or "colored" by the prevailing notion of American as a white identity. He says: 

He would not bleach his Negro soul in a flood of white Americanism, for he knows that Negro blood has a message for the world. He simply wishes to make it possible for a man to be both a Negro and an American, without being cursed and spit upon by his fellows, without having the doors of Opportunity closed roughly in his face. 

This problem of double consciousness does still occur today. Consider the NFL. 68% of the players are black while 31% are white. And yet, only 13% of the head coaches are black. With a league/business/group that has a majority of black employees, one would think that this trend would extend to higher ranking positions. But this is not the case. Progress has been made, but this is an enduring example of racial bias. It is an example of belonging (playing in the NFL) but not belonging (limited opportunities for higher positions such as being a head coach). This stems from the archaic, prevailing notion that those higher positions have been/are more suited for whites. To the extent that racism is still a part of our society, African Americans continue to confront the "double consciousness" described by Du Bois.