Defining Shakespeare's dramas is a near-impossible task. His plays were, in great part, responsible for changing the entire theater experience, presenting the drama as it had never been seen before.
Shakespeare often used stories that had been told by others: Romeo and Juliet uses a theme that...
...appeared in the fourth century in a Greek tale...
But it was the way that Shakespeare treated his subjects, used the language and related to things that audience cared about that changed the face of drama forever. Marlowe and Kyd are playwrights that were also popular at the time, but Shakespeare is most often associated with refining the genre—using models from Aristotle and Horace, but making the dramatic art his own.
"The Bard" did not write about religious themes, as earlier English plays did, but spoke to more secular issues. In Macbeth, the question of valid regicide is considered: Elizabethans believed (in keeping with the Chain of Being) that God ordained who would be king. Macbeth's murder of Duncan was considered by the audience to be a mortal sin. It raised the question for Elizabethans—something they would have struggled with in the past—as to if regicide was ever justified: for in the play, Macbeth is a tyrant.
With Hamlet, Elizabethans would have had strong opinions about Gertrude's remarriage to her brother-in-law, which was then seen as an incestuous union—something Hamlet laments in his soliloquy in Act One, scene two:
...She married. O, most wicked speed, to post
With such dexterity to incestuous sheets! (159-160)
Shakespeare also addressed the supernatural: Elizabethans deeply believed in witches, ghosts and omens. The ghost of Hamlet's father comes from the grave to demand that his death be avenged: Hamlet struggles with whether the Ghost or good or evil, one reason he delays in killing Claudius.
Shakespeare also changed the way the world looks at fairies, as seen in A Midsummer Night's Dream. Until that time, fairies were perceived as evil and malicious. Not so with Shakespeare's fairies, for he...
...eradicates the old beliefs and creates the new fairy himself...fairies now dwell in places of fancy instead of horrifying abodes.
Lastly, Shakespeare used elements of drama (soliloquies, asides, etc.) to enable his audience to identify with his characters. Shakespeare's mastery with the language of the day drew his audience into the frustrations of the evil Richard III, the sorrow and anger of Hamlet for his father's murder, and Macbeth's struggle between being moral or killing his King—he admits that heaven will bewail the death of such a great man, and that he has little reason to follow such a path:
I have no spur
To prick the sides of my intent, but only
Vaulting ambition, which o'erleaps itself
And falls on the other— (I.vii.25-28)
Shakespeare, in his comedies, tragedies and dramas is able to capture the imagination of his audience: they believe there is a battle raging on the stage, righteously hate Richard III, admire Brutus even as he kills Julius Caesar, sympathize with Hamlet's heartache, and laugh as Puck declares:
Lord, what fools these mortals be!A Midsummer Night’s Dream (III.ii.110–115).
Shakespeare's writing, use of dramatic techniques and, most of all, his ability to connect to his audience, with...
...the interplay of human character and motive...
...are characteristics of his works that make them relevant to audiences hundreds of years after they were first presented on stage.