A stock exchange is a LEGAL entity that is meant to bring together buyers and sellers. Though they were created by legal rulings at the SEC, they create and enforce their own rules (i.e., they are self-regulated).

A stock broker is a professional person or entity that transacts - i.e., buys and sells securities - on behalf of their verified and trusted customers.

Trades happen both on stock exchanges and 'off-exchanges' - and have different rules and regulations through each. Exchanges have protections - like the NBBO - that are supposed to keep purchase prices within limits.