The key thing to understand is that what you have is actually essentially two different applications, and those applications talk to eachother via network requests. For what you are describing, generally your Node server would expose an API, which you then trigger via AJAX requests from the in-browser JS code.
You can read up on exposing API methods in Node here: http://blog.modulus.io/nodejs-and-express-create-rest-api
For the client side of things, you should read about $.ajax
and its arguments.