Who represents whom in "The Crucible", and why?
Arthur Miller wrote this play as a commentary on the Red Scare that happened in the United States where people engaged in a "witch hunt" of communists. Many people were brought before the government and questioned about being a communist. They were brought in if anyone even suspected they were a communist; they were ratted out by fellow workers, friends, and brought in for any tiny communist-esque slants in their work. So, in "The Crucible" the courts and the judges would represent Senator McCarthy (who led the red scare in the senate) and the others in government who questioned people and blacklisted them from working, just like Danforth and others jailed "witches". Proctor would represent any rebellion movement against the red scare, and the voice of reason and logic against mass hysteria that often gets squelched in the craziness that accompanies any type of witch hunt. Miller used Proctor and others like him to symbolize the ulterior motives that many people had for labelling people as witches, or as communists; for example, Abby was jealous of Elizabeth and so labelled her a witch. Most of the accused in "The Crucible" can be tied back to greed, hatred, insecurities, grudges and hatred-not to actual witchcraft. Miller also felt that many of the accusations of communism were motivated by politics, fear, and personal vendettas that were given a "just" playing ground.
Those are just a couple ideas; I hope they helped.
Although The Crucible was written because Miller considered many of the emotional and power dynamics of the Salem Witch Trials similar to the persecution of communists and fellow travelers by Joseph McCarthy and the House Un-American Activities Committee (HUAC), it is not actually a "roman à clef" ("story with a key") in which historical figures are intended to have a one-to-one correspondence with Miller's contemporaries.
While Miller did take some poetic license, combining characters and slightly altering their biographies, in fact, the characters in his play are all based on real characters who actually lived in Salem.
The correspondences we find between Miller's world and that of Salem have more to do with group identities, with Parris, Hathorne, and Hale parallel to people holding strongly anti-communist beliefs, the girls similar to those who were caught up in the McCarthyite hysteria, and Corey, Proctor, and Nurse parallel to innocent victims persecuted by McCarthy.