Add Real-Time Chat to My App

Add a real-time chat feature to my app using WebSockets.

Framework: [Express + Socket.io, Django Channels, FastAPI WebSockets, etc.]

I want:
1. Users can send messages and see them appear instantly
2. All connected users see new messages in real-time (no page refresh)
3. Show a simple "user is typing..." indicator
4. Handle what happens when someone disconnects

Keep it simple: just the core chat functionality. Show me:
- The server-side WebSocket setup
- The client-side JavaScript to connect and send/receive messages
- How messages get broadcast to everyone

I'm learning WebSockets for the first time, so explain each part simply.