When Atticus says, “Simply because we were licked a hundred years before we started is no reason for us not to try to win” (101) in To Kill a...

Atticus is explaining to his children that even though he is unlikely to win the trial, he should fight anyway.

When Mrs. Dubose dies, Atticus explains to his children that she was fighting against a terrible addiction.  She was addicted to morphine and dying, but she wanted to die free from the addiction.  This was why Atticus sent his children to read to her.  He wanted them to see a different definition of courage than what they would expect.

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. You rarely win, but sometimes you do. (Ch. 11)

Mrs. Dubose fought hard in a seemingly unwinnable fight.  It was largely a mental fight.  She had to win against herself.  As impossible as it was, she succeeded.  Atticus wanted to set her up as a role model for his children because he knew that they were able to face a fight themselves as the town of Maycomb turned against him and them.

Earlier, Atticus explains to Scout that he is defending a Negro, Tom Robinson.  She does not understand why he is doing something that everyone in the town considers a terrible betrayal.

“Atticus, are we going to win it?”

“No, honey.”

“Then why—”

“Simply because we were licked a hundred years before we started is no reason for us not to try to win,” Atticus said. (Ch. 9)

For Atticus, not being able to win the case is not a reason not to try.  This is something he is trying to reinforce with the example of Mrs. Dubose.  He will continue to fight because it is the right thing to do, and because he has to stand up for what he believes in.  Tom Robinson deserves his best efforts, regardless of his race.