An Exchange is simply an institution that allows you to buy and sell crypto.
Exchanges allow people to open accounts, transfer fiat or crypto into the account, buy and sell crypto, and transfer fiat and crypto out. Of course there are fees associated with "trades" (crypto buys and sells), and gains made on exchanges are required to be reported on tax returns.
There are other institutions not considered exchanges, such as PayPal, where you can buy BTC and use the BTC to make purchases. You are required to report gains made after selling BTC purchased or used to purchase items on those platforms.
Every exchange is different, and not all exchanges trade all cryptos. Also, with a lack of clarity from the SEC, as of June 2023, it is impossible to know which cryptos are considered securities and which are considered assets, with the exception of Bitcoin, which is considered an asset. This has caused a great deal of mayhem in the crypto market lately, and makes it very difficult for the consumer to know which exchange(s) to use.