This was a question I posed to my own students in our Renaissance chapter. I received varying answers that mainly came straight from the book. Is Shakespeare still relavent to high school students today with Twlight being the success that it is?
My first response and the most simple one regarding the Bard's relavance would be, "No. He is not really relavant to kids today." But he is relevant to scholars and students of literature.
The conventional reason for his continuing relavance to literature might go like this: Shakespeare was extremely productive and represents a connection between an old world - of whose culture we know very little intimately - and the world we live in today.
Not only did he create dozens of literary works, he is also said to have coined over 10,000 new words and phrases in English, which gives you some insight into his productivity.
Though the quality of his works is uneven, and not always good, Shakespeare is the cornerstone of the so-called Western Cannon, perhaps simply because no one else offers a wide and varied enough body of work to really fill this role.
Personally, I think that there are two basic answers to the question "Why is Shakespeare considered to be so great?". One - He wrote quite a bit of good material, some of the most compelling and intriguing plays and poems in English. His work, some of it, is truly great and, maybe, he wrote more great works than anyone else. (Most writers who are considered great have only produced a single masterpiece.). Shakespeare's talent and volume of work afford us with a wealth of ideas produced by arguably the greatest imagination working in literature ever.
Two - a lot of people don't really understand Shakespeare's plays and they are afraid to say they don't like them. It's the same kind of pride-and-shame response that makes people say the James Joyce is the greatest writer of the 20th century.
To be slightly cynical I do think that one of the reasons why Shakespeare is so enduring and popular is the sheer amount of written material he produced (though even this is debatable). I do think that the plays and poetry he leaves us with are excellent and raise so many universal issues, but I see the same quality in other playwrights of the time such as Marlowe. Perhaps Shakespeare's ability to outlive other contenders have made him what he is today. Controversial, I know!
Don't forget that the issues Shakespeare deals with are still very relevant today...regardless of nationality, gender, ethnicity, and economic status. He deals with incest, love, envy, greed, ambition, family relationships, hatred, ghosts, guilt, and every economic level from royal heads of state to the most impoverished. The language is beautiful, and he speaks to us about all the issues we are interested in...including the paranormal and human fears (ie, vampires?)
I pose a similar question to my 9th graders when teaching "Romero and Juliet." I too receive varying answers, but invarilably we get into a discussion of Shakespeare's command of rhythm and his mastery of words. In an effort to modernize the rhythm, I link Shakespeare to rappers such as Eminem in their methods of emphasizing different syllables in words and phrases, etc in order to create the rhythm they are intending in the writing. I also point out to students how Shakespeare invented many new words, which poets and lyricists often do today as well. Students usually bring up how word meanings have changed over time according to how they are currently used in pop culture. High school students may not immediately see the relationship, but Shakespeare is definitely still relevant in literature, especially when studying rhythm and word choice (which you can do in any novel, poem ,etc, including Twilight.)