What does Atticus mean when he says, "Simply because we were licked a hundred years before we started is no reason for us not to try to win"?

Atticus wants Scout to understand that just because something is difficult doesn't mean you should not do it if it is the right thing to do.

Atticus tells Scout that there is no way he is going to win the case when she gets into a fight with Cecil Jacobs after the boy insulted him.  Scout doesn’t understand why people are so upset with her father.  He explains to her that he is defending Tom Robinson.  Tom was accused of raping a white woman.  He is a black man.  To the people of Maycomb, that makes him guilty.

Atticus feels that just because he was assigned the Robinson case does not mean he should not put his best efforts into it.  He wants to try, even though he knows he will not win.  It is a matter of personal pride to him.

Scout tells Atticus that he sounds like Cousin Ike Finch, who likes to rehash the Civil War.

“It’s different this time,” he said. “This time we aren’t fighting the Yankees, we’re fighting our friends. But remember this, no matter how bitter things get, they’re still our friends and this is still our home.” (Ch. 9) 

Atticus knows that the people of Maycomb will not want to confront the racial realities of the case, but he feels that it is his job to show them.  It is common in Maycomb to assume that a black person is guilty, no matter what.  No one in Maycomb is willing to consider otherwise.  Atticus wants to give them a chance. 

When Mrs. Dubose dies, Atticus uses her battle with her morphine addiction as an example of moral courage.  He wants his children to remember that sometimes you have to do things that are very difficult, and perhaps even impossible.  The description he gives of Mrs. Dubose fighting her addiction could just as easily apply to his taking the Tom Robinson case. 

I wanted you to see what real courage is, instead of getting the idea that courage is a man with a gun in his hand. It’s when you know you’re licked before you begin but you begin anyway and you see it through no matter what. (Ch. 11)

Atticus realizes that it will be very hard to win Tom Robinson's case, because of Maycomb's deeply ingrained racism.  However, he wants to try, because it is the right thing to do for Tom Robinson and for Maycomb. He does not get Tom Robinson acquitted, but he does get the jury to deliberate longer than any other Maycomb jury has.  That shows progress.