The dream deferred in Langston Hughes's famous poem is the dream of an America where there will be true equality among the races and where African Americans, specifically, will no longer be discriminated against and oppressed. Hughes's final question, "or does it explode?", carries the implication that when a people's freedom is denied for years and years, the possible result is violence of some kind.
In the play A Raisin in the Sun, the Younger family's dream is to move to a better neighborhood. The attempt is made, by a so-called "welcoming committee" of residents, to prevent them from doing so, purely out of racist motives. This is the specific dream that has been deferred, although the family decide to defy the bigots and take their chances.
The broader dream that has been put off or deferred is, of course, that of equality and freedom from oppression in general, as implied in Hughes's poem. Each member of the Younger family understandably feels stymied in his or her personal life as a result of this deferral. Walter wonders why he can only be a chauffeur for a rich white man. Beneatha feels trapped because she is both black and a woman, and the rest of the family have little understanding of why she feels frustrated. Even Asagai, who as one born in Africa is in some sense less affected by the mindset that results from systematic discrimination, seems a bit amused by Beneatha's inward struggle.
Mrs. Younger is the one whose steadfastness holds the family together. Though her dream is the one that's been deferred the longest, it's chiefly through her guidance that the family resist the forces ranged against them and intend to fight to make the dream a reality.