What does iambic pentameter symbolize in Shakespeare's plays?
There are a couple of ways that your question can be interpreted; so I'll answer it in two ways.
1. First, Shakespeare wrote his plays in blank verse featuring iambic pentameter because that was the style of the day. Think of it as a way for an author to show off--and it really is quite impressive if one thinks about it. There are very few authors who can create characters and plots as rich as Shakespeare's and write their lines in a consistent meter.
2. Secondly (I think that this might be what you are asking), when Shakespeare's characters speak in verse (iambic pentameter), they are usually the noble (aristocratic) characters, and their speech represents their high culture and position in society. If you simply look at one of Shakespeare's plays, you can often tell when the commoners are speaking because their lines will go from margin to margin (this is true, too, of nobles who are acting like commoners--whether they're involved in evil schemes, losing their minds, or are drunk!). In contrast, Shakespeare's other characters' lines should sound and look different to you--they should sound "sing-songy" and should look like poetry with uneven lengths.
A good example of this is from Othello. When Iago is speaking to his peers or to those in position of authority over them, his speech is in verse, but when he is plotting and talking to Roderigo (especially at the play's beginning), his lines are not in iambic pentameter--this represents the bawdy nature of his speech and, in truth, the baseness of his character.
Iambic pentameter does not symbolize anything. It is a rhythm. It was introduced by playwrights like Christopher Marlowe and the University Wits.
The rhythm is quite simple---de DUM, de DUM, de DUM, de DUM, de DUM. It alternates the stress. It was used because it most closely reflects the number of syllables we get said in a single breath. In other words, it most closely mirrors everyday English speech rhythms.
Ask any actor and they will tell you that verse is easier to learn than prose. This was important during the Elizabethan period because actors had to keep approximately 30-40 plays in their head. If someone requested to see a play they had not performed for a while, there was no brush up rehearsal.
Shakespeare used iambic pentameter as a norm upon which he could change the rhythm much like a jazz muscian.
It is true that for the most part, high born or noble characters use verse, and servants and lower class characters use prose but this is not always true. For example, in Romeo and Juliet, Mercutio, a kinsman of the Prince, speaks more prose than verse. In his case, he is speaking informally, he is chatting with friends.
A character like Caliban in The Tempest speaks verse although he would be considered a low born character. In his case, he learned to speak from Prospero.
There are no strict rules as for who speaks what. Iambic pentameter is a rhythm, nothing more complicated than that.
Another example of the difference between the use of iambic pentameter and prose for a specific purpose in a play is seen in Hamlet. Hamlet, as the prince of Denmark, speaks in iambic pentameter throughout the play, EXCEPT when he is acting mad. The use of regular, prose speech was one way that Shakespeare could further highlight the difference in Hamlet's "crazy act" character from his normal, sane character. Shakespeare's audience would have heard the difference in the meter of the speeches and when combined with the subject matter of the speech and perhaps his physical actions could discern that Hamlet was putting on his act. He speaks in prose when he is around Polonius, King Claudius, his friends Rosencrantz and Guildenstern, and with the grave-diggers. These are all of the situations where Hamlet is trying to deceive those around him in the hopes that they may lower their guard and reveal the truth about Claudius' murder of Hamlet's father, King Hamlet.