Yes, Shakespeare was a real person. We know this because his name shows up on several legal documents for the purchase of land not to mention there are quite a few other contemporary references to him as well. The theory that Shakespeare's plays were written by someone else is rooted in the fact that Shakespeare had much less traditional schooling than many other playwrights of the time so it is thought that he couldn't have written the caliber of plays that he did. Having said that, there isn't definitive proof that the plays couldn't have been written by someone else but Shakespeare is widely accepted as the author. Source, Norton Anthology of Shakespeare