In Chapter 9 of Harper Lee's To Kill a Mockingbird, Atticus's reference to being "licked a hundred years before" is a reference to slavery and pre-Civil War days.
Prior to Atticus's reference, Scout had just been verbally attacked by Cecil Jacobs, who demanded to know why her "daddy defended niggers." Cecil's comment reflects the deep-seated racist views of the South. Feeling confused because she thinks Cecil has just pointed out that her father is doing a bad thing, Scout asks her father if he is defending a Negro, and Atticus's response is that all lawyers defend Negroes.
Atticus further continues to explain that he is defending Tom Robinson, who is "a member of Calpurnia's church," and his family members are known by Calpurnia to be "clean-living folks." When Scout next asks why he is defending Robinson, Atticus replies that he "couldn't hold up [his] head in town" if he didn't. Atticus's speech to Scout and responses to her questions indicate that Atticus is defending Robinson because he believes in Robinson's innocence and disbelieves in the town's racist views. Hence, by the time Atticus argues that, though he knows they'll lose the case, just because they "were licked a hundred years before" the case started, doesn't mean they shouldn't try to win, we know Atticus is referring to the racist views that gave birth to slavery and started the Civil War.
The novel is set during President Roosevelt's days of the New Deal, during the Great Depression, which was in the 1930s. Therefore, a 100 years prior to when the book is set would be the 1830s, days when slavery still reigned in the South and 30 years prior to the start of the Civil War. It was racism that gave birth to slavery and racism that kept African Americans from receiving the justice and freedom due to them as American citizens. Racism also gave birth to the Civil War because the South felt that maintaining an economy that relied on slave labor and their racist views were worthy of fighting to the death for. Though the South lost the war, Atticus is well aware that the fight to end racism is a losing battle, a battle in which those who disbelieved in racism "were licked," meaning defeated, and had been licked for the past hundred years. Hence, all in all, Atticus is using his reference to slavery and pre-Civil War days to show that those who are just, merciful, and non-racist have been being defeated by those who are racist for a very long time.
However, Scout misinterprets his meaning. She relates what he is saying to Cousin Ike Finch, Maycomb's only living Confederate veteran, who still complains of the South having lost to the Yankees its authority to uphold racist views through slavery. Atticus is too respectful a person to tell Scout that Cousin Ike's views are racist; he instead only tells her that, this time, the fight against racism is a fight between friends.