How did the New York Stock Exchange work?

How exactly does the New York Stock Exchange function? I'm curious about the mechanisms behind its operation and how it enables the buying and selling of stocks. What are the key roles played by various participants, such as traders, brokers, and investors? Additionally, how are trades executed, and how does the exchange ensure fairness and transparency in the process? Lastly, how does the NYSE stay relevant in today's digital age with the rise of cryptocurrencies and alternative trading platforms?