Stock exchange is basically a marketplace where people go to buy and sell stocks. The screens typically display the current market price of individual stocks so people know roughly how much they should expect to pay or receive. They can also display information about past trends (have stocks been going up or down), significant news events, etc. that might be helpful for deciding when to trade stocks.
To trade stocks, you need to find someone who’s willing to buy those stocks from you (or sell them to you), negotiate the price, and make the transaction. You can do this yourself, but unless you want to treat it like a job, it isn’t worth the effort. It’s much easier to find an agent or broker who will do this on your behalf. Then you just need to tell them what to buy/sell, and they can gather everyone’s orders and fulfill them all at the same time. Nowadays most brokers are online and most transactions are made electronically, so the crowded room with brokers shouting numbers is becoming a thing of the past.
Stock exchanges generally are open only during business hours, so technically there aren’t people working in stock exchanges 24/7. However it is possible to trade stocks even when the markets are closed, and some agents will work extended hours if their customers need it. With online and electronic brokers, their servers and computer systems can operate 24/7.
Nowadays those trading floors are much less crowded since any meaningful transactions can be done online or over the phone. Their continued existence borders on skeuomorphism.
Trades can be done at any point in the day. Delineation of business hours is a method of recording and measuring value over time (Opening day price, closing day price). Functionally, business hours is when there are the largest number of active traders so the market price of stocks is going to be more stable. After hours, with fewer traders operating, there’s going to be more volatility since those fewer transactions can lead to some abrupt changes in price.
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