Exchanges in the crypto sense are covered in full under Cryptocurrency Exchanges, which describes how these platforms work, the main types and how they are regulated. This page gives a short definition and explains why the bare word "exchange" can be ambiguous.
In crypto, an exchange is a marketplace where users trade digital assets for other digital assets or for government money. The two main models are the centralized exchange, a company that holds customers' funds and matches their orders on its own books, and the decentralized exchange, where trades settle directly between wallets through smart contracts. Centralized platforms usually require identity checks and, in many countries, a licence; decentralized ones often use pools of tokens rather than order books.
Outside crypto, "exchange" has older meanings: a regulated stock or futures market, the act of converting one currency into another at an exchange rate, or a swap of goods. Crypto glossaries use the plain word as shorthand for trading platforms, which is why this site files the topic under the fuller name. Holding coins on any exchange carries custody risk if the operator fails or is hacked.