Actually, nobody is sure exactly what day Shakespeare was born. I guess his parents didn't realize how famous he was going to be—because the only date we have in writing is his baptism date, April 26, 1564.
Still, it's traditional to celebrate Shakespeare's birthday on April 23, which is St. George's Day and is also the date that he died (in 1616).
You probably already know that many people consider Shakespeare to be the greatest writer in the English language and the greatest playwright in ANY language. The things he has written (a few with other authors) include 38 plays, 154 sonnets, two long narrative poems, and several other poems.
To celebrate the day, read a sonnet or two, or watch a movie version of one of Shakespeare's plays. How about A Midsummer Night's Dream (released in 1999, rated PG-13), Much Ado About Nothing (released in 1993, rated PG-13), or Romeo and Juliet (released in 1996, rated PG-13)? Note that all Shakespeare plays (and therefore movies based on those plays) have some measure of adult content, so check out parent advisories!
Check out this list of words and phrases originated or at least popularized by Shakespeare.