In Chapter 15 of To Kill a Mockingbird, the tone begins to change.  It becomes more serious and Scout, the narrator, begins to see injustice in society.  Scout refers to the time as a nightmare.  This nightmare refers to the trial of Tom Robinson and the implications of it for Atticus and his family.  

One night, the sheriff and a group of men from the area come to the Finch house.  They ask to speak to Atticus.  Sheriff Tate warns Atticus of possible violence because Tom is being moved to the jail in Maycomb.  Atticus dismisses this idea:

"Don't be foolish, Heck," Atticus said. "This is Maycomb."

The next evening, Atticus sits outside of the jail where Tom is.  An angry mob approaches.  Scout, Jem, and Dill have snuck out of the house and are hiding near the jail.  They run to Atticus when they see the mob.  Scout speaks kindly to Mr. Cunningham, the leader of the group.  Her kindness causes him to rethink his threats to harm Tom, and he orders the group to leave.

The trial then begins.  Tom's trial is filled with bias because he is an African American man accused of a crime against a white woman.  Scout and Jem witness the trial, which is like a nightmare to them as they see the injustice firsthand.

The nightmare that descends upon Jem and Scout--and everyone else in Maycomb--is the trial of Tom Robinson. After spending "a week of peace together" with Dill after he had run away from home, bad things begin to happen on the weekend before the trial is scheduled to begin. Atticus receives word that Tom is about to be moved to the local jail and that there may be trouble from "that Old Sarum bunch." Sure enough, a lynch party appears at the jail just as the three children descend upon it to spy upon Atticus. The children know something is not right, but they don't realize that lives are in danger until the next morning. When the children sneak into the courtroom and observe the very adult facts of the rape trial, they become emotionally involved. Dill is brought to tears by Tom's treatment at the hands of the prosecutor, and Jem cries following the unjust guilty verdict. Bob threatens Atticus afterward, and even Tom's death does not end the children's nightmare, since Bob Ewell has plans for them on the following Halloween night. For Jem--and Bob--the nightmare ends tragically, though it is Boo Radley who appears as if in a dream to save the kids from Bob's murderous hands.

The nightmare that is referred to in Chapter 15 begins when a group of men gather outside the house, waiting for Atticus to come out. Jem, Dill, and Scout gather at the window to watch the scene unfold. The men eventually disperse when confronted with Atticus' unflinching equanimity and the realization that there are children inside.

Despite Atticus's reassurances to the contrary, Jem is convinced that the group of men were a gang there to harm Atticus. The nightmare Scout refers to as "upon us" at the beginning of the chapter is partly the threat of harm being inflicted upon Atticus, Jem and Scout's beloved father. Jem also makes a connection between the group of men who had gathered outside the house and the infamous Klu Klux Klan.

The children's fear that Atticus will be harmed is heightened later in Chapter 15 when Atticus is approached at the county jail, this time not by a group of men familiar to the children but by a group of "sullen-looking, sleepy-eyed" strangers. The strangers are there to take Tom Robinson, and Atticus seems certain to be hurt until Scout's innocent intervention.

The nightmare that descends upon the children is primarily the nightmare that something bad will happen to Atticus, who for them is the personification of security and goodness. However, the nightmare is also, in a broader sense, the nightmare of the growing tide of racism and the accompanying threat of violence that escalates as the trial of Tom Robinson approaches.

The nightmare unfolding around the children’s safe and innocent life is the racism the children will face throughout the arrest and trial of Tom Robinson.  Scout and Jem first confront the racism of the town when they follow Atticus to the jail.  Atticus’ intention is to protect Tom from a lynch mob, and when the children arrive at the jailhouse, the mob is threatening and angry.  Scout, in her innocence, points out Mr. Cunningham in the mob.  Mr. Cunningham represents the racism present in the average Maycomb resident, and this scene allows Jem and Scout to see that this hatred is predominant in people they know and the everyday events of Maycomb.  

Throughout the rest of the novel, Scout learns about hypocrisy through the teachings of her teacher about Hitler’s discrimination of Jews and the outlandish hypocrisy of the missionary circle raising money for a tribe in Africa but who won’t help their neighboring black community.  She also learns about Dolphus Raymond and the discrimination he must endure to live the way he wants.  Jem and Scout are also harassed by their classmates at school for Atticus defending Tom.  Even Mrs. Dubose yells at the children and calls their father a derogatory name. 

The children are living in a nightmare because they are being ostracized and bullied by their friends and acquaintances.  They are no longer living the dream-like state of children but are living the reality of how racism can affect them and those around them. 

At the beginning of Chapter 15, Scout says, "A nightmare was upon us." (Lee 193) The nightmare Scout was referring to was the mob scene that takes place later on in the chapter. Harper Lee foreshadows this dangerous scene at the beginning of the chapter. Atticus goes by himself to sit outside of Tom Robinson's cell the night before the trial. Jem, Scout, and Dill follow Atticus and hide next to Tyndal's Hardware Store, and watch as the four cars arrive. The Old Sarum bunch gets out of their cars and surrounds Atticus. The mob plans on harming Tom Robinson. Scout runs into the group to greet Atticus and spots Walter Cunningham. After multiple attempts at "small talk," Scout gains Walter's attention. Walter realizes the precarious situation that Atticus is in, and tells the mob to leave. Scout unknowingly saves both her father and Tom Robinson from possible harm. The next day, Atticus teaches Scout and Jem a valuable lesson in mob mentality. 

To Kill a Mockingbird, is one of the most beloved works of literature. Harper Lee has such a way of putting you right there in the middle of the story. 

There are a few different meanings of what the nightmare is to the children. In this part of the book, several things are taking place. The trial of Tom Robinson is all the town can think or talk about. The people of Maycomb are so ready to find the man guilty. Atticus is trying hard to keep all of the hatred away from Jem and Scout. When the mob approaches the jail, Jem, Scout and Dill are afraid for Atticus. This is the first time they had seen their neighbors act this way. It is also a nightmare for them to learn the attitude of the people against a black man. Jem and Scout are losing the innocence they once took for granted. The other nightmare for the two of them is what they have to endure with Mr. Ewell. Their lives are almost taken away from them, if it weren't for Boo.

The nightmare for Jem and Scout, unfortunately is reality. Atticus can't save them from the harsh realities of the world. All he can do is be the best influence on them and teach them how to treat people, which is exactly what he does.

The nightmare refers to the racist element in Maycomb. Atticus agreed to take Tom's case and this is in a town where racial tension and segregation still exist. The nightmare is that, since Atticus has defied the social code of Maycomb by defending an African-American, he (and his family) will have to deal with the backlash from that racist element.

While Tom is in Maycomb jail, prior to his trial, Mr. Tate and others come to Atticus' front lawn because they're concerned a mob might go after Tom. There is also concern, particularly from Jem (in Chapter 15) that Atticus himself could be in danger. These concerns are confirmed when a mob does indeed show up at the jail while Atticus is standing guard. Scout and Jem interrupt the mob. Jem is defiant and will not leave, despite Atticus' demands to do so. Scout directly addresses Walter Cunningham, Sr., and through her innocence and kind words about Walter, Jr., Walter Sr. convinces the mob to disperse. Ironically, Scout had thought that this was the same group who voiced concern on Atticus' front lawn. "There was a smell of stale whiskey and pigpen about, and when I glanced around I discovered that these men were strangers." (Chapter 15)

The nightmare temporarily goes away but returns when Tom is wrongfully convicted and when Bob Ewell continues to harass Atticus, Helen, and eventually the children. In general, the nightmare is this racist element and it also becomes Bob Ewell's personal vendetta against Atticus and what he represents.

Scout is alluding the the real start of the trial.  Though the trial is still in the future, this really marks when Scout begins to lose some of her innocence as part of seeing Atticus wrapped up in the trial.  Plus, she will see a more malevolent side of Maycomb than she has ever seen before.

The nightmare in Ch. 15, of course, is the Old Sarum bunch that come to lynch Tom Robinson, who is being held in the local jail.  Of course,  Atticus chooses to sit outside of the jail and make sure that Tom is safe.

The kids sneak out and go to visit Atticus just as the mob arrives.

Atticus attempts to reason with the mob, but, in the end, it is Scout that saves the day.  This is important because she is able to make Mr. Cunningham stand in Atticus's shoes, so to speak, for a moment.  He sees Atticus as a father and friend rather than just someone standing in their way.  Their anger is defused and they leave.

Scout saves the day and shows the reader that she was embodied one of the novel's main themes - empathy.

However, this scene serves to foreshadow the rest of the nightmare that is to come - the trial, the injustice,  the racism, and, ultimately, Bob Ewell's revenge.

Just to add a couple thoughts to the previous answer, the nightmare could also be how the family is now thrust front and center into the debate surrounding the trail.  Since Atticus is flying in face of Maycomb's normal social conventions, simply lynching Tom Robinson and attaining mob justice, he is bringing down all kinds of trouble on to his family.  This can range from simple insults and dirty looks all the way up to the revenge Bob Ewell seeks on Atticus and his children.

The "nightmare that was upon us" is how Scout describes, in Ch. 15, the events that led up to Atticus' confrontation with the townsmen while he was protecting Tom in jail.  For the first time, Scout and Jem witness the hatred of the mob and the horrors of racism, and see their father in a vulnerable, and perhaps deadly, position.