With a broker, an organization sets the price and fee that they’re willing to sell for. They either hold the funds or work with a network of other brokers in order to keep sufficient supply. Brokers will likely keep prices similar to that of others in the same markets in order to keep demand for their services. Coinbase is an example of a digital currency broker.
With an exchange, there are multiple buyers and sellers that are placing offers to buy and sell simultaneously. Buyers can choose to buy at any price, but the order will not be completed until a seller agrees to the transaction. The equilibrium price on these exchanges are set as the last agreed on price between buyers and sellers.
Both brokers and centralized exchanges have existed in traditional finance. In a centralized exchange, a governing organization creates the exchange and sets rules around how much information participants must share, withdrawal limits, and what can be traded on the exchange. Binance is an example of a centralized exchange.
A new type of exchange being built for the digital currency world is the decentralized exchange. Instead of having a governing organization, the rules of the exchange are set by a smart contract. Bitshares (BTS) and Ether Delta are examples of current decentralized exchanges and 0x (ZRX) is building a protocol (like an API) that would allow any company to build exchange functionality into their Decentralized App (DApp)