You create a webSocket or socket.io connection from your client to your server and your server is the webSocket or socket.io server. Once that connection is established, you can then freely send data either way across the connection, from client to server or from server to client.
This type of architecture is ideal for sending data from your server to a web page to dynamically update the web page as new data arrives.
webSocket is the base transport. socket.io is a layer on top of webSocket that adds a bunch of useful features such as auto-reconnect and structured messages. You can use either from a browser. webSocket support is built-in to the browser. If you want to use the additional features of socket.io, then you include the socket.io client library in your web page.
Here's a listing of some of the additional features socket.io offers over a plain webSocket: Moving from socket.io to raw websockets?.