What is the relevance of Shakespeare's plays for a modern audience?
One of the reasons that Shakespeare is said to be a writer of "timeless" literature is that his topics and, especially his characters, speak to a modern audience just as forcefully as they did when the Bard was writing. If we look to Shakespeare's characters, we find that some of the men have the best of intentions, but are brought down by the evil of those around them, as seen with Hamlet, the "haunted" prince of Denmark in the play by the same name, and Brutus in Julius Caesar. Hamlet wants to avenge his father's murder while avoiding the forfeiture of his eternal soul (as killing a king—Claudius—was considered at the time to be a mortal sin). Brutus' part in the assassination of Caesar can be seen as an act of treason, but begged the question of the Elizabethan audience as to whether it was lawful to depose or kill a king (or queen) if that person was a danger to the state. It was for this reason that Brutus joins Cassius—Brutus fears what Caesar will do to Rome if he is declared "king," while Cassius is simply jealous and resentful toward Caesar, wanting him dead simply for Cassius' personal gratification. Brutus sacrifices everything for the noble purpose of protecting Rome, which he loves more than his own life.
Our society is not without its villains, and we can see such in the person of Macbeth, who kills his king to become the ruler of Scotland. Shakespeare describes tyrants other than Macbeth: King Lear who is a tyrannical father; and, Duke Frederick who grasps at the reins of power by usurping his own brother and exiling him. (There are many more examples in Shakespeare's "historical" plays.)
Shakespeare's women provide a colorful tapestry of characters we can identify with: Lady Macbeth is a schemer and willing partner in her husband's murder of Duncan; Gertrude marries her brother-in-law, committing incest (as the Elizabethans saw it) after her husband, Old Hamlet, dies—she is a women trying to survive in a man's world; Ophelia, Hamlet's sweetheart, is destroyed by the machinations of this man's world, losing her mind as she is toyed with and heartbroken; she drowns and the men around her (especially the Church) insist she committed suicide when a branch she sat on broke and she fell into the water. We are mesmerized by the Weird Sisters in Macbeth who play with Macbeth's mind and induce him to sacrifice his soul to be king; and we are further haunted by the witches' queen, Hecate, who wants nothing more than to destroy Macbeth motivated by pure evil. The audience sees innocence destroyed by the fighting of families in Romeo and Juliet.
Even in the face of tragedy and violence, Shakespeare also speaks to common foolishness that abounds when we don't pay attention, try to manipulate those around us, or simply lose sight of what is important. These things serve to entertain us when the comedy of human error is displayed in plays such as A Midsummer Night's Dream and The Taming of the Shrew.
Shakespeare's characters, and the themes that drive the plots, are rife with examples of the evil of some, the foolishness of others, and those lucky enough to recognize their mistakes before it is too late. We can identify with Shakespeare because the people he described beginning in the late 1400s, are much like those we see in the news and in our neighborhoods. The beauty of his writing allows us to further see the genius of the man, and his plays are still performed to appreciative audiences today.