I think that you will find different potential points to a question like this. I can only suggest that you take in what you get here along with reading or familiarizing yourself with the work, Eyes on the Prize. This work does a great job of analyzing the movement and identifying important parts and figures in the movement.
I would say that one of the most important parts to know about the Civil Rights Movement is to examine the leadership of the movement. Figures like Dr. King and Malcolm X and providing the contrast between both leaders brings out the complexity of the movement and the existential quality of how different leaders attract support from different segments of the population. In the end, too often, the movement is seen as a blanket and monist element. Examining the leadership of the movement and who they attracted is vitally important to bringing out the complex nature and the nuanced substance of the period and the movement that gave voice to it. Who attracted which types of people and how those people represented the what the movement was all about is representative of the importance of the movement. I think that the study of leadership is an important part in the study of the Civil Rights Movement in the 1960s.