Intro
The textual history of Shakespeare's plays is a total mess. We don't actually have any of his plays passed down to us in his own hand. In fact, we only have texts of Shakespeare's plays at all because people in his company—actors and others—transcribed his works and preserved them in documents that are known as Quarto 1, Quarto 2, and the Folio.
As you might expect, there can be some very important differences as you move from one version of his works to the next. Let's look at this example from King Lear; it's from the last scene of the play, when Cordelia, Lear's daughter, dies in Lear's arms. King Lear's been through so much crap by this point that the death of his youngest daughter is the last straw, and he dies. Here's the same scene from the Quarto 1 and Folio versions of the play.
Quote
Quarto 1 Text
Lear: And my poore foole is hangd, no, no life, why should a
dog, a horse, a rat [have] life and thou no breath at all, O thou wilt
come no more, neuer, neuer, neuer, pray you vndo this button,
thanke you sir, O, o, o, o.
Edgar: He faints my Lord, my Lord.
Lear: Breake hart, I prethe breake.
Edgar: Look vp my Lord.
Kent: Vex not his ghost, O let him passe,
He hates him that would vpon the wracke,
Of this tough world stretch him out longer.
Folio Text
Lear: And my poor Foole is hang'd: no, no, no life?
Why should a Dog, a Horse, a Rat haue life,
And thou no breath at all? Thou'lt come no more,
Neuer, neuer, neuer, neuer, neuer.
Pray you, vndo this button. Thanke you Sir.
Do you see this? Looke on her? Looke her lips,Looke there, looke there. [He dies.]
Edgar: He faints, my lord, my lord.
Kent: Breake, heart, I prythee breake.
Edgar: Look vp, my lord.
Kent: Vex not his ghost. O, let him passe, he hates him,
That would vpon the rack of this tough world
Stretch him out longer.
Analysis
There are some pretty significant differences between the Quarto version of King Lear and the Folio version. One of the main differences is that in the Folio text, Lear has a couple of extra lines: "Do you see this? Look on her! Look, her lips. / Look there, look there." It's as if in his final moment of life, he sees his dead daughter coming back to life. These lines are not in the Quarto version.
Another big difference is that the last words that are attributed to Lear in the Quarto version ("Breake hart, I prethe breake") are given to Kent, Lear's loyal friend, in the Folio text.
There are also a number of smaller differences between the Quarto and the Folio text. Lear's "No, no, life" in the Quarto becomes "No, no, no life?" in the Folio text. His "Neuer, neuer, neuer" becomes "Neuer, neuer, neuer, neuer, neuer" in the Folio. And all those "Os" are nowhere to be found in the Folio version.
The point is that these differences, even if they seem small at first glance, are totally significant. They change the way we view Lear, and they change the way we view Kent. Thematically, there are big differences: if Lear actually sees something like Cordelia coming back to life, that's a major thematic point as well as a major development of Lear's character.
Textual Critics have a tough job to do here: they have to figure out which of these two versions of Shakespeare's play is the more authoritative. Generally, Textual Critics agree that the Folio version of Lear is the more authoritative of the two versions available, but it's the kind of thing they have to look at line by line by line.