The ugly facts of life in particular that Atticus reminded Jem of had to do with the immoral actions of people.
All of we readers, as well as Jem, Scout and Dill could tell that the case should have gone in favor of Tom Robinson. Atticus never comes out and blatantly says it but we all know the reason he didn't get off was because the jury was prejudice against Tom's skin color.
Atticus did point out that the jury was out for a long time and that one man took some considerable wearing down if they needed to come up with a unanimous verdict.
A fact of life that this reveals is that our prejudice can lead us to do something that is wrong or immoral, like sentencing an innocent man to life in prison or worse, death. That prejudice was murder. Who now stood trial for Tom's death? No one. That's another ugly fact of life.
The trial is over, Tom has been convicted, and Bob Ewell has spit in Atticus' face and threatened his life. In Chapter 23, Jem is upset about all of these things, but he is particularly worried for his father. When Jem declares that juries should be done away with, Atticus defends the court system. Atticus also explains to Jem that "nothing in your life has interfered with your reasoning process." Atticus explained that every man sees things in different ways, and that Jem is a reasonable person. The jurors, and the men who came to lynch Tom at the jail, were not reasonable, Atticus explained.
"There's something in our world that makes men lose their heads. In our courts, when it's a white man's word against a black man's, the white man always wins. They're ugly, but those are the facts of life."