A submit button usually is for submitting a form. Submitting a form is sending a request to the server and refreshing the page. So in your server side script you could read the posted values and show them in the resulting page (you don't need javascript for this).
If you don't want to redirect you could handle the submit event and cancel the default submission:
var values = [];
$(function() {
$('#id_of_form').submit(function() {
// get the entered value:
var userInput = $(':input[name=userInput]').val();
// add the current value to the list
values.push(userInput);
// show the values
alert(values.join(", "));
// cancel the default submission
return false;
});
});