Stream-of-consciousness writing usually involves the narrator speaking of whatever comes into their head, without any predictable or planned structure or topic. It is as if you set your thoughts into a stream, and allow the currents to take you wherever they may go. It is an interesting form of writing because it can reveal quite about about a person's character; where their mind rambles in their journey reveals how their thoughts work, what they worry about most, and can expose vulnerabilities and interesting revelations that would be kept under check in more structured, planned writing.
In "The Love Song of J. Alfred Prufrock," Eliot uses stream-of-consciousness writing to reveal the character traits of the narrator, Prufrock. The central thought that his character, Prufrock, is pondering is whether or not to ask a loved one an important question, presumably a marriage proposal or some other feelings-related question. However, as he ponders if, when and how to do this, his mind rambles along a lot of other pathways, which reveal important character traits. As he speaks of parties, he reveals his distaste for their futility; as he speaks of women, he reveals his intimidation of them; as he speaks of great men and literature, he reveals his disconnect with their heroes; as he speaks of himself, he reveals his incredible insecurity and longing for meaning in life. Every time his mind wanders off on its stream of thought, however, it always loops back to the stressful issue at hand--his desire to communicate something important to his loved one.
"The Love Song of J. Alfred Prufrock" is a great example of stream-of-consciousness writing and how it can be used to reveal character in a creative way. I hope that those thoughts helped; good luck!