Quince: "We will have a prologue, and it shall be written in eight and six." Bottom: "Nea, make it two more... eight and eight." What exactly did they mean by those numbers?

A debate in Shakespeare's A Midsummer Night's Dream over which type of sonnet they should use for the play within the play. This "play within the play" was titled "Pyramus and Thisbe". The first, mentioned by Quince, is referring to the octave (eight lines) and the sestet (six lines) of the fourteen line sonnet. This is the normal structure used for prologues.

As for what form the "eight and eight" refer to, I need to think about that. Obviously it's a sixteen line form of poetry, with two octaves.

Alternating lines of eight and six syllables (rhyming ABAB)made up the traditional English ballad metre, found for example in “Chevy Chase”:

“The Perse owt of Northombarlande
A vowe to God mayd he ....”

Uniform eight-syllable lines (rhyming AABB) were favoured in romances, e.g. Chaucer’s translation of the Romance of the Rose:

“Without abode, knele down anon
And you shal have absolucion.”

So Quince fancies a simple unsophisticated folksy ballad metre, while Bottom thinks that a more polished courtly romantic style is more appropriate.

See this webpage (page 638):
http://rsnz.natlib.govt.nz/volume/rsnz_43/rsnz_43_00_008400.html

Thank you TabbyTom!!! I was just doing that from memory :)

Ahh.. The perks/curse of an old English major lol

And which, of course, does Shakespeare have his character write?

The prologue turns out to be in the heroic style - iambic pentameters (or ten and ten):

If we offend, it is with our good will ....

Well, I'm quite impressed by the knowledge of literary devices displayed here. A silver pen then for you.