Stock traders can trade on their own account, called proprietary trading or self-directed trading, or through an agent authorized to buy and sell on the owner's behalf. That agent is referred to as a stockbroker. Agents are paid a commission for performing the trade. Proprietary or self-directed traders who use online brokerages (e.g., Fidelity, Interactive Brokers, Schwab, tastytrade) benefit from commission-free trades.
Major stock exchanges have market makers who help limit price variation (volatility) by buying and selling a particular company's shares on their own behalf and also on behalf of other clients.