Why are William Shakespeare's plays important today ?
it is a very general question. as correctly written above, "important" is not quite clear. however, shakespeare has had a huge imapct on the english language. for the same reason people go back to read historical documents like anne franks diary, the ancient egyptian heiroglyphics, others look back at shakespeare's work with the same interest and look for his contribution in english history.
the themes that are discussed in his plays are universal, since each has somehting that people fo all ages can relate to.
Finally, shakespeare was an entertainer. He has clearly done a good job if people still want to go and read his works and enjoy watching his plays.
hope this helps.
Although not Shakespeare's biggest fan (as I'm sure many others could claim to be), after having studied his work through various stages of my academic life - from highschool through to university, I do appreciate the relevency of themes he explores (whether intentional or not). I believe that everyone, even younger generations can find something in Shakespeare in which they can identify with - be it:
Teenagers - forbidden love - tragic end (Romeo & J.), sibling rivalry (King Lear), ambition - corruption (Macbeth), manipulation - jealousy (Othello), gender uncertainty (Twelfth Night), revenge (The Tempest), etc.
This is one of the reasons why I believe that Shakespeare is still important today, together with various other reasons. One other is perhaps the fact that he was first and foremost an entertainer and helped shape the modern day theatre that many of us enjoy today.