Why are Shakespeare's plays/ works still relevant today?

One reason Shakespeare remains relevant is because he created characters anyone can relate to. He developed such a wide variety of characters that it is almost impossible not to find one who appeals to a reader. He wrote about teenagers in love and old men nearing the end of their lives. He depicted love and grief in a very real, human way.
Another reason Shakespeare remains relevant is because of his stories themselves. We still read fairy tales and remember their characters and morals because they teach lessons. Just as with fairy tales, we remember the lessons Shakespeare teaches in his stories. In Romeo and Juliet, for example, we learn the pitfalls of "fast love" and unbridled passion/ violence.  
The themes in Shakespeare's work reflect the present day, too. Today, racism and sexism exist. Shakespeare explores these themes in plays like The Merchant of Venice. We can look at these themes in Shakespeare and apply them to our own lives.