In many instances, you will want to send data from the R server to the JS client. Here is a very simple example:

library(shiny)
runApp(
  list(
    ui = fluidPage(
      tags$script(
        "Shiny.addCustomMessageHandler('message', function(params) { alert(params); });"  
      ),
      actionButton("btn","Press Me")
    ),
    server = function(input, output, session) {
      observeEvent(input$btn,{
        randomNumber <- runif(1,0,100)
        session$sendCustomMessage("message",list(paste0(randomNumber," is a random number!")))
      })
    }
  )
)

The workhorses here are the session$sendCustomMessage function in R and the Shiny.addCustomMessageHandler function in javascript.

The session$sendCustomMessage function lets you send parameters from R to a javascript function, and Shiny.addCustomMessageHandler let's you define the javascript function that accepts the parameters from R.

Note: Lists are converted to JSON when they are passed from R to javascript

In some instances, you will want to send data from JS client to the R server. Here is a basic example using javascript's Shiny.onInputChange function:

library(shiny)
runApp(
  list(
    ui = fluidPage(
      # create password input
      HTML('<input type="password" id="passwordInput">'),
      # use jquery to write function that sends value to
      # server when changed
      tags$script(
        '$("#passwordInput").on("change",function() {
          Shiny.onInputChange("myInput",this.value);
        })'
      ),
      # show password
      verbatimTextOutput("test")
    ),
    server = function(input, output, session) {
      # read in then show password
      output$test <- renderPrint(
        input$myInput
      )
    }
  )
)

Here we create a password input with id passwordInput. We add a Javascript function on the UI that reacts to changes in passwordInput, and sends the value to the server using Shiny.onInputChange.

Shiny.onInputChange takes two parameters, a name for the input$*name*, plus a value for input$*name*

Then you can use input$*name* like any other Shiny input.