Power and authority in this play come in two basic forms - moral and social/economic.
The social and economic power of the Younger family is thoroughly demonstrated in its limitations in the play. The check the family receives, which facilitates the drama of the play, presents a rare opportunity for the family - a chance at financial empowerment. The choices that this check allows the family to make, once articulated, are indications of the family's actual dis-empowerment.
The money is to be used for escape from the family's current situation.
This is true for all of the ways they would like to spend the money, from medical school for Beneatha (allowing her to claim some earning power, some dignity of position, and some intellectual respect), a liquor store for Walter (allowing him to take some ownership in his life and take charge of himself, his family's well-being, and express his manhood in ways he cannot as a chauffeur), and buying a house (to escape the blighted and drab neighborhood they now live in).
All of these dreams relate to the idea of social and economic power. The family now has the temporary power in the insurance check to do something positive.
Importantly, the flush of power the family enjoys is contrasted to that of the home owners group that attempts to buy out the Younger family before they move into their new neighborhood. This group wields power as a matter of course, unlike the Younger family who have a short-lived exposure to financial power.
The other significant type of power in the play is moral power and moral authority. Many of the play's dynamics feed into the thematic idea that true power is found in the ability to maintain priorities, to recognize one's place in a family structure (and in a culture) and find a way to say "yes" to that place.
Walter becomes the head of the family as his mother had hoped he would only when he learns this moral lesson and takes a position of moral leadership. Through much of the play Mama stands in this position and her authority is derived from her moral power.