The Civil rights movement began to splinter due to the different approaches that African-American leaders wanted to take in order to cause change.
For example, Martin Luther King Jr. was an advocate for nonviolent forms of protests in order to incite change. This includes marches, sit-ins, boycotts, wade-ins, etc. On the other hand, there were groups like the Black Panther Party that felt that African-American citizens had the right to use violence as a means of self defense. This group, lead by people like Malcolm X, disagreed with nonviolent protests. Malcolm X and the Black Panthers felt that this did not create change and only left African-American citizens powerless against white aggression/violence.