Ibsen is often thought of as the founder of realist drama. Other playwrights who wrote in this genre include Chekhov, Strindberg, Shaw, and O'Neill. In realist drama, the characters talk in a close approximation of everyday speech. The vast majority or modern plays, TV shows, and movies, are written in a similar style, though most fail to rise to the same level of social critique.
Realism shouldn't be confused with its Naturalism. Though the two styles were being developed around the same time, they have some significant differences. Basically, Naturalism was just a lot more hard core about representing everyday life exactly as it is. Naturalist plays wanted to show a slice of life exactly as it is. Characters might talk on and on about nothing in particular and the plays might have no obvious climax – just like most days are for most people.
Realism, however, is unafraid to be a little unrealistic. Look at A Doll's House. Sure the characters talk in a generally conversational way, but the plot is obviously and unapologetically contrived. There are melodramatic devices like secret revealing letters. The doorbell rings at convenient times, bringing trouble for Nora. People enter and exit just when Ibsen needs to move on to the next scene and bring on new ideas. This wasn't a bad thing to Ibsen. His goal was to examine ideas, to challenge individuals to really think about their society, not to present photographic reality. A Doll's House is widely considered to be one of the prime examples of realism.