When we talk about "tone," we're referring to the author's and/or the play's attitude toward its subject matter.
At the beginning of A Midsummer Night's Dream, the tone is pretty dark, wouldn't you say? After all, Hermia faces the death penalty or life as a nun if she doesn't obey her father and marry the man of his choosing. This suggests a bleak outlook, don't you think?
Still, this darkness quickly gives way to a lighthearted tone that reveals Shakespeare's sense of humor about the pitfalls of love. Case in point: When the young lovers (some of whom have been drugged by Oberon's magic love potion) go chasing each other around the wood, falling in and out of love at the drop of a hat (or the drop of some magic love juice), Shakespeare pokes fun at how erratic and foolish we can all be when it comes to romance. Just ask Titania, who falls head over heels in love and spends all her time lavishing her love and attention on a (literal) jackass.
P.S. If you're feeling all emo and want to read one of Shakespeare's darker, more cynical plays, check out Measure for Measure, where a wannabe nun faces the death penalty if she refuses to sleep with a corrupt deputy.