Henrick Ibsen is the father of the modern realistic theater and A Doll's House was the most discussed play of the late 1800's.
There are many different kinds of theater -- or illusions -- because what is happening on the stage -- the "space" -- is not real.
A play is an imitiation of an action though sometimes the action is so compelling that for several hours we believe that the action on the stage is really happening and is real life.
This illusion that real life has been created on stage is the staple of many movies and TV dramas and is what most people think of as going to see a play even though more money is probably spent on shows like CATS and PHANTOM OF THE OPERA --- both of which are more "fantastic" than "real."
So what is a realistic play? To answer this question we must divide the term realism into several parts:
Form: The illusion of the 4th wall.
What is real life and real drama?
Sneak over to your neighbor's house every night at 8:00 and peek through the living room window. What you see and hear is real life -- though depending on where you live you may have to wait anywhere from several days to several months before anything "dramatic" takes place.
The lives of ordinary middle class people like you and me in real houses in real time is the stuff of realistic drama. But who would wait weeks in a theater for something meaningful to occur? So the playwright foreshortens everything and re-arranges a few details so something dramatic and exciting unfolds right away. Even the dialog is changed. Real conversation is never as interesting or as literate as a play script. So while the actors appear to be having a "real" conversation , the dialog is actually very contrived.
In fact everything is contrived! When we sit in the theater and see a living room set --- the 4th wall has been removed so we can see into the room -- we believe that the action is real, but everything from the pre-arranged props and the timely exits and entrances has been fabricated. So realistic theater is just as much of an illusion as a fantasy is --- except that the laws of physics operate according to common sense -- no Peter Pans fly across the stage and all actions are motivated: when an actor moves there must be a reason for the movement -- answer the phone, get a drink, ect. When a scene is first rehearsed, it is "blocked" -- every position and movement carefully worked out so as to create the "illusion of reality." Finally as for the people in realistic theater, there are no Kings and Queens as characters, no castles or Titanic ships as backdrop --- just ordinary human beings like you and me struggling with life's problems and conflicts.
Historically
Realism as a dramatic art form dates from the late 19th Century --- specifically with the plays of Ibsen like A Doll's House and Hedda Gabler.
Almost all theater before Ibsen was non-realistic.
Think about it: Greek tragedy -- Kings and Queens in an open universal space with a Chorus chanting and commenting on the action. Shakespeare? Romance and Melodrama with spectacular scenery and heros and heroines more noble, handsome and o'er- reaching than you or I could ever be.
Ibsen wanted to address the life and the concerns of ordinary middle-class citizens who were quickly becoming the majority class politically and economically. By 1900 what Kings and Queens were left in Europe were figureheads with little political power. World War I would be the aristocracy's last stand. There was no longer a need to see kings and queens on stage. Politically they no longer had any power. Morally they were bankrupt.
Subject-matter
What were the concerns of the middle class? Besides a realistic set, common dialogue (no speeches or solilioquies) and motivated action, Ibsen wanted to make the stage a vehicle for the discussion of modern-day problems. (Talk radio does this today with ordinary citizens talking about same sex marriage, the death penalty, abortion. etc.)
A Doll's House and Hedda Gabler deal with women's liberation. An Enemy of the People takes up political corruption not very much different from the Washington shenanagans of today. This aspect of realism is know as "the problem play" --- a drama which exposes a specific social problem. TV realism does this as well -- but usually with a morally simplistic point-of-view: At the end the bad guys are in jail and justice previals. But "real" life is not often so simple --- especially with complex social and ethical concerns. A good problem/realistic play is more troubling. Justice is not so easily gained. As with real life sometimes there are no good or easy solutions.