I'll veer away a little from the philosophical discussion of modern drama because the other posts have covered that very thoroughly. Another common aspect of modern drama is a focus on stage direction. Modern playwrights often include background information and detailed stage directions (movement of characters, stage setup, even light and sound effects). Sometimes they provide detailed costume instructions or descriptions. This focus on the sets and background information differs from Renaissance drama, which often provides minimal stage direction.
A good example of this is Miller's The Crucible (or many of his plays for that matter). He produces a tragic hero and many other classical effects but then structures his play in a modern style.
In the twentieth century, realism to extreme, naturalism, expressionism, and symbolism characterized drama. Such as play as "The Iceman Cometh" by Eugene O'Neill is noted for its dark realism. Written in 1939, this play was not performed until 1946 after World War II when O'Neill felt the public was ready for the despair in this play. In 1956 O'Neill's "Long Day's Journey into Night" was performed; this play presents an almost morbid pessimism as none of the characters have any noble characteristics. Often O'Neill has been classified as a naturalist; that is, he emphasizes the more sordid and pessimistic aspects of life.
In the mid-twentieth century, expressionism burgeoned with its use of minimal scenery, telegraphic dialogue, and characters portrayed as types, rather than individuals. Expressionism was used by Tennessee Williams in his "A Glass Menagerie." Such an expressionistic play approaces truth not by aping reality, but by transforming it into illusion. The use of music, for example, is employed as memory. Light, too, plays an important role as many of the characters' movements are formalized and mechanical, as well. Williams's play is viewed through a screen to aid in the perception of the characters as more type than whole. With the continuing effects of World War II, other works that were expressionistic were produced to express the meaningless of human existence. From expressionism there was a natural transition to the theatre of the absurd in such plays as Samuel Beckett's "Waiting for Godot" and Eugene Ionesco's "Bald Soprano." This theatre of the absurd at times assaulted its audience with movement and sound and cruelty such as in Edward Albee's "Who's Afraid of Virginia Woolf" whose dialogues are brillantly abusive and cruel. Such a play is marked by wit, irony, and satire. Existentialism was a natural progression from the theatre of the absurd with its need for expressing the psychological effects of knowing that life is absurd and that one must makes his/her own existen
In the late twentieth century, a witty surrealism emerged with brillant word play, stark settings, austere language in sparse dialogue came about with playwrights such as America's Sam Shepard and David Marmet in "Buried Child," and "Glengarrry Glen Ross," plays characterized by stark settings, meaningful silences, powerful streaks of menace, and outbursts of real or implied violence. Certainly, modern drama has opened up many avenues for playwrights.
This question will open a great deal of discussion and I am not sure you will have a definitive answer. Perhaps, you will have to piece together ideas to derive your own understanding. I look at modern drama from a thematic perspective. Part of what defines modern drama for me is an emphasis on experiences and predicaments that have applicability to as many people as possible. Modern drama speaks loudly and lucidly to multiple parties, and can articulate struggle and redemption in a manner that makes it understandable to all in the modern setting. Its relevancy is effective in real time. For example, a reason I consider Beckett's Waiting for Godot modern drama because it speaks to a condition of paralysis that can apply to human beings, as a whole. The Crucible is an example of modern drama because it speaks to the vision of the tyranny of the community and the hypocrisy that it compels within individuals. This definition of modern drama can encompass works that apply to a particular culture of individuals. I consider Angels in America, a play that deals with thematic elements about homosexuality, an example of modern drama because it speaks to how one deals with death and the fear of it. Fences might be a work that is applicable to the African American predicament but it speaks to a larger conception of dreams and conflict within families. For me, modern drama has to speak to issues where there is a level of connection that can be evident to as many individuals as possible as something new in the human predicament is revealed.