How did Shakespeare's plays get handed down to us?

There are two basic ways.  During Shakespeare's life,
anyone who could get a handwritten copy of a play could print it.  There were no
copyright laws at the time, and, since Shakespeare was a popular playwright, anyone who
could get a copy of, say, a script used to prompt the actors during a show could print
it in order to make money.  These printed copies of plays were called quartos because
the sheets of printing paper were folded four times.  Eighteen of Shakespeare's plays
exist in one or more quarto versions.

The other way the
plays have come down to us is what is called the First Folio (published in 1623).  After
Shakespeare's death in 1616, two of Shakespeare's fellow actors printed 36 of the 38
plays attributed to the Bard in a large book.  Some plays (such as The
Winter's Tale) only exist in the First Folio.  Others, like
Hamlet or King Lear, exist in both the First
Folio and in one or more quarto versions.  Sometimes the differences between the
versions are relatively minor and sometimes they are very significant.  King
Lear is a good example of the latter; some editions of Shakespeare's complete
works print two versions of this play because the First Quarto is so different from
others.

So, when modern editors want to publish a copy of
one of Shakespeare's plays, they have to consult all of the versions out there and
decide what they want to do about any differences.  In addition, they have to make
decisions about punctuation, confusing words, and problems like that.  Unfortunately,
with the exception of one page of writing, as far as we know, none of the plays in
Shakespeare's own writing have survived.  Gladly, we do have the quartos and the First
Folio.