Since you specifically asked about the use of blank verse in A Midsummer Night's Dream, I have moved this question.
Essentially there are three worlds in the play: the world of the court, the world of the workers, and the world of fairy land. In both in the court and fairyland, blank verse is spoken while Bottom and his mates speak prose.
Since both the court and fairy land are formal worlds, the language is formal. Bottom and his mates are workers. They speak prose to show there are of a lower class. It also shows that these men are friends and they speak an informal language. Even when Titiana speaks verse to him, Bottom speaks in prose.
Shakespeare often used prose to show lower class characters against the higher class ones. This is not always the case. For example the play Much Ado About Nothing is about 60% prose.
As stated by cldbentley, blank verse is a rhythm. From the time it was first used by Christopher Marlowe, it became the standard for Elizabethan theatre. (It is easier to memorize and most closely resembles everyday English speech rhythms.)
Shakespeare used blank verse a great deal throughout his writings. Blank verse is a form of poetry containing lines that do not rhyme. Although they are unrhymed, these lines do consist of iambic pentameter, which is five stressed beat in every line with every other syllable being stressed. Often, it is easier to really notice the stressing of syllables when they are read aloud than when they are read silently.
In Shakespeare's plays, characters who spoke iambic pentameter were usually important. Minor or insignificant characters often did not speak in verse at all.