You could easily write a major BOOK about how he has influenced our lives, regardless of where in the world you live. The following are some bullet points, just in case you are NOT writing a book and merely an essay. :)
*Characterizational depth. He offered a glimpse into the dark side of human nature on a consistent basis.
*Introduced recognizeable archetypes, symbols, icons
*stabilized a literary genre
*wrote for the common man (as well as the aristocracy)
*words from his plays are used around the world as part of the common lexicon (rotten in Denmark, parting is such sweet sorrow, all the word's a stage)
*involved the elements and setting as part of the characterization
*Portrayed women as having a certain strength (moreso than others during his day and age)
*developed characters with depth and breadth