Shakespeare was baptized on April 26, 1564. For symmetry, many prefer to believe he was born on April 23, which is the date in 1616 that he died.
None of Shakespeare's plays were published during his lifetime, although he published a collection of sonnets (1609) and others published pirated editions of a few of the plays. The first collected edition of the plays, known as the "First Folio," was published by friends in 1623.
See the essay on this sculpture in Outdoor Monuments of Manhattan: A Historical Guide, and the episode in Monuments of Manhattan (with another web page here).