When you ask about the aspects of "Tragedy" utilized by Shakespeare, it seems necessary to consult the Greek philosopher Aristotle, who originally defined the term in his treatise Poetics. Though Aristotle was writing about the Greek tragedies (such as Oedipus Rex) of his day, we assume that Shakespeare used Aristotle's definition and the tragedies of the Ancient Greek theatre as his models for tragedies such as Hamlet, Romeo and Juliet, Macbeth, Othello, Julius Caesar, Antony and Cleopatra and King Lear.
Here is Aristotle's definition of a Tragedy, as reprinted by Enotes:
Tragedy is a representation of a serious, complete action which has magnitude, in embellished speech, . . .[represented] by people acting and not by narration; accomplishing by means of pity and terror the catharsis of such emotions.
Of these elements, many would argue that the ones that make Shakespeare's tragedies excellent or superior are his use of "embellished speech" or, in Shakespeare's case, iambic pentameter, and the "accomplish[ment] by means of pity and terror the catharsis [for the resolution] of such emotions."
Shakespeare's language, the words and phrases that he invented that we have incorporated as part of our everyday speech -- the many, many famous lines spoken by characters such as Hamlet, King Lear, Romeo, Othello, Iago, Juliet, Macbeth, and Lady Macbeth -- show that, even though Shakespeare wrote his plays over 400 years ago, their "embellished speech" is of such superiority and excellence, that some of it is common everyday parlance here in the 21st century.
The "pity and terror" leading to "catharsis" which is experienced as a suitable resolution by the audience viewing a tragedy by Shakespeare can be attested to today as well. The superiority and excellence of Shakespeare's tragedies have been demonstrated in that, as Enotes says, they have:
. . .been translated into many languages and performed on amateur and professional stages throughout the world. Radio, television, and film versions of the plays in English, German, Russian, French, and Japanese have been heard and seen by millions of people. . . . Novelists and dramatists such as Charles Dickens, Bertolt Brecht, William Faulkner, and Tom Stoppard, [have been] inspired by Shakespeare’s plots, characters, and poetry. . .. A large and flourishing Shakespeare industry exists in England, America, Japan, and Germany, giving evidence of the playwright’s popularity among scholars and the general public alike.
This amazing and ongoing world-wide popularity of the works of this playwright from 16th century England, is a testament to what a superior and moving experience these plays are for audiences all the world over. People from any culture, any background, find meaning in experiencing the journeys of characters in Shakespeare's tragedies. Characters such as Hamlet, Macbeth and King Lear provide the audience the necessary "pity and terror" that lead to "catharis." They have done so for over 400 years, and we can only assume that they will continue to do so into future generations.
For more on Aristotle's Poetics and his definition of Tragedy, as well as more on William Shakespeare the Dramatist and words/phrases he coined, please follow the links below.