The Civil Rights Movement

Harper Lee worked on To Kill a Mockingbird throughout the late 1950s, amid the beginning of the Civil Rights Movement. In this era, African-Americans in the United States fought to overcome legal discrimination, segregation, disenfranchisement (lack of voting rights), and other types of discrimination by using protest tactics such as sit-ins, civil disobedience, boycotts, marches, etc. Most major legal battles were won in the mid-1960s, but generation of activists not unlike Lee herself were gaining major momentum through the 1950s.

The late 1950s would have been particularly tense in towns like Monroeville, Alabama (Harper Lee's hometown) in the Deep South, where racial tensions ran deep and many of the most violent, race-related crimes took place.

It's important to note that while issues of gender identity and equality are evident in To Kill a Mockingbird, the Civil Rights Movement didn't focus on women's rights until 1963, after Mockingbird was published.

Timeline:
1954: Supreme Court decision Brown vs. Board of Education rules that segregation in public schools is unconstitutional.
1955: In Mississippi, Emmett Till (a 14-year-old black boy) allegedly whistles at a white woman and is brutally murdered for it. The two white men accused are acquitted of the crime.
1955: In Montgomery, Alabama, Rosa Parks (a black woman) refuses to give up her seat for a white man on a bus and is arrested. African-Americans boycott city buses and they are eventually desegregated.
1957: In Little Rock, Arkansas, President Eisenhower sends federal troops to protect nine black students attempting to attend an all-white public high school.
1960: Four black students sit-in at a Woolworth's lunch counter in Greensboro, North Carolina.

Links:
CNN: Basic timeline of the movement.
Watson.org: Great overview of important protest events and methods.
Dennis M. Simon: Very clear overview of the most important turning points.