Three patterns, often combined
Event: server pushes, client reacts. Notifications, price updates, presence — anything the user did not ask for in this moment. Command: client tells server to do something; server may or may not respond. Sending a chat message, updating profile, joining a room. Pub/Sub: client explicitly subscribes to topics; server only sends messages for subscribed topics. Useful when one connection multiplexes many independent feeds.
Most apps use all three
A trading platform: pub/sub for market data (subscribe to symbols), commands for placing orders, events for fills. A chat app: pub/sub for joining rooms, commands for sending messages, events for typing/presence. The patterns compose; they are not exclusive.