Why do the early printings of Shakespeare's plays disagree, and what is the First Folio?

The early printings disagree because plays were not first published as polished books. They reached print from different sources, some good and some questionable, and a play's text could change over the years through revisions, cuts and performance. Hamlet is the famous case: three early versions survive, the First Quarto (1603), the Second Quarto (1604) and the First Folio (1623), and each has lines the others lack.

They also differ in size: the First Quarto has 15,983 words, the Second Quarto 28,628 and the Folio 27,602. For most of his career, Shakespeare's plays were sold as cheap quartos, which could be bought unbound for 6d and were meant to be read until worn out. Seven years after his death, two fellow actors, John Heminges and Henry Condell, collected 36 plays in a prestigious folio, the First Folio of 1623. Eighteen of those plays, including The Tempest and Macbeth, were not known to have been printed before.

Heminges and Condell called the earlier prints stol'n and surreptitious copies. In 1909 the bibliographers Alfred Pollard and W. W. Greg turned that remark into a theory of bad quartos, unauthorised texts that actors in minor roles may have reconstructed from memory. Today that theory is contested, and the truth may be more mixed: texts can differ because a play was revised or shortened over the years.