Shakespeare has his characters speak in prose when they are of the lower social classes. You will notice that all of the "mechanicals" - Bottom, Quince, Snout, Snug, Flute, Starvling - all speak in prose. It is partly Shakespeare's way of showing that these characters are not sophisticated or educated. They are rooted in the mundane, the ordinary; the "prosaic," if you will. Poetry is heightened speech, the language of love and philosophy and great drama, and these characters tread upon the ground.
By contrast, the men and women of the court all speak in verse. This is appropriate to their status as upper-class characters, whose discourse is about more high-minded things.
However, it is also important that you see that frequently Shakespeare has characters (such as Puck and the Fairy) speak in a verse which is more song-like, in eight-syllable lines instead of the ten of iambic pentameter. This is to suggest the fanciful, folk-tale like quality of the fairy world.