Market makers as firms and their role in providing liquidity are covered under market maker, and the other side of each trade is described under market taker. This page explains the pairing of "maker" and "taker" as it appears in exchange fee schedules.
On an exchange that uses an order book, every trade matches two orders. The maker is the party whose order was already resting in the book, usually a limit order placed away from the current price. The taker is the party whose incoming order fills immediately against it, typically a market order or a limit order priced to cross the spread. A professional market-making firm is mostly a maker, but any trader becomes a maker whenever their order waits in the book.
The distinction matters because many exchanges charge makers lower fees, or even pay them a rebate, while charging takers more. The aim is to reward those who add liquidity and narrow the bid-ask spread. Some exchanges offer "post-only" orders that are cancelled rather than filled as a taker, so traders can be sure of maker pricing. Whether a trade counts as maker or taker depends only on the timing of the orders, not on whether it is a buy or a sell.